A unified neurofuzzy model for classification

This work proposes a unified neurofuzzy modelling scheme. To begin with, the initial fuzzy base construction method is based on fuzzy clustering utilising a Gaussian mixture model (GMM) combined with the analysis of covariance (ANOVA) decomposition in order to obtain more compact univariate and bivariate membership functions over the subspaces of the input features. The mean and covariance of the Gaussian membership functions are found by the expectation maximisation (EM) algorithm with the merit of revealing the underlying density distribution of system inputs. The resultant set of membership functions forms the basis of the generalised fuzzy model (GFM) inference engine. The model structure and parameters of this neurofuzzy model are identified via the supervised subspace orthogonal least square (OLS) learning. Finally, instead of providing deterministic class label as model output by convention, a logistic regression model is applied to present the classifier’s output, in which the sigmoid type of logistic transfer function scales the outputs of the neurofuzzy model to the class probability. Experimental validation results are presented to demonstrate the effectiveness of the proposed neurofuzzy modelling scheme.

Data assimilation with correlated observation errors: experiments with a 1-D shallow water model

Remote sensing observations often have correlated errors, but the correlations are typically ignored in data assimilation for numerical weather prediction. The assumption of zero correlations is often used with data thinning methods, resulting in a loss of information. As operational centres move towards higher-resolution forecasting, there is a requirement to retain data providing detail on appropriate scales. Thus an alternative approach to dealing with observation error correlations is needed. In this article, we consider several approaches to approximating observation error correlation matrices: diagonal approximations, eigendecomposition approximations and Markov matrices. These approximations are applied in incremental variational assimilation experiments with a 1-D shallow water model using synthetic observations. Our experiments quantify analysis accuracy in comparison with a reference or ‘truth’ trajectory, as well as with analyses using the ‘true’ observation error covariance matrix. We show that it is often better to include an approximate correlation structure in the observation error covariance matrix than to incorrectly assume error independence. Furthermore, by choosing a suitable matrix approximation, it is feasible and computationally cheap to include error correlation structure in a variational data assimilation algorithm.

Representativity error for temperature and humidity using the Met Office high-resolution model

The observation-error covariance matrix used in data assimilation contains contributions from instrument errors, representativity errors and errors introduced by the approximated observation operator. Forward model errors arise when the observation operator does not correctly model the observations or when observations can resolve spatial scales that the model cannot. Previous work to estimate the observation-error covariance matrix for particular observing instruments has shown that it contains signifcant correlations. In particular, correlations for humidity data are more significant than those for temperature. However it is not known what proportion of these correlations can be attributed to the representativity errors. In this article we apply an existing method for calculating representativity error, previously applied to an idealised system, to NWP data. We calculate horizontal errors of representativity for temperature and humidity using data from the Met Office high-resolution UK variable resolution model. Our results show that errors of representativity are correlated and more significant for specific humidity than temperature. We also find that representativity error varies with height. This suggests that the assimilation scheme may be improved if these errors are explicitly included in a data assimilation scheme. This article is published with the permission of the Controller of HMSO and the Queen's Printer for Scotland.

Fast identification algorithms for Gaussian process model

A class identification algorithms is introduced for Gaussian process(GP)models.The fundamental approach is to propose a new kernel function which leads to a covariance matrix with low rank,a property that is consequently exploited for computational efficiency for both model parameter estimation and model predictions.The objective of either maximizing the marginal likelihood or the Kullback–Leibler (K–L) divergence between the estimated output probability density function(pdf)and the true pdf has been used as respective cost functions.For each cost function,an efficient coordinate descent algorithm is proposed to estimate the kernel parameters using a one dimensional derivative free search, and noise variance using a fast gradient descent algorithm. Numerical examples are included to demonstrate the effectiveness of the new identification approaches.

Estimating correlated observation error statistics using an ensemble transform Kalman filter

For certain observing types, such as those that are remotely sensed, the observation errors are correlated and these correlations are state- and time-dependent. In this work, we develop a method for diagnosing and incorporating spatially correlated and time-dependent observation error in an ensemble data assimilation system. The method combines an ensemble transform Kalman filter with a method that uses statistical averages of background and analysis innovations to provide an estimate of the observation error covariance matrix. To evaluate the performance of the method, we perform identical twin experiments using the Lorenz ’96 and Kuramoto-Sivashinsky models. Using our approach, a good approximation to the true observation error covariance can be recovered in cases where the initial estimate of the error covariance is incorrect. Spatial observation error covariances where the length scale of the true covariance changes slowly in time can also be captured. We find that using the estimated correlated observation error in the assimilation improves the analysis.

Satellite-supported flood forecasting in river networks: a real case study

Satellite-based (e.g., Synthetic Aperture Radar [SAR]) water level observations (WLOs) of the floodplain can be sequentially assimilated into a hydrodynamic model to decrease forecast uncertainty. This has the potential to keep the forecast on track, so providing an Earth Observation (EO) based flood forecast system. However, the operational applicability of such a system for floods developed over river networks requires further testing. One of the promising techniques for assimilation in this field is the family of ensemble Kalman (EnKF) filters. These filters use a limited-size ensemble representation of the forecast error covariance matrix. This representation tends to develop spurious correlations as the forecast-assimilation cycle proceeds, which is a further complication for dealing with floods in either urban areas or river junctions in rural environments. Here we evaluate the assimilation of WLOs obtained from a sequence of real SAR overpasses (the X-band COSMO-Skymed constellation) in a case study. We show that a direct application of a global Ensemble Transform Kalman Filter (ETKF) suffers from filter divergence caused by spurious correlations. However, a spatially-based filter localization provides a substantial moderation in the development of the forecast error covariance matrix, directly improving the forecast and also making it possible to further benefit from a simultaneous online inflow error estimation and correction. Additionally, we propose and evaluate a novel along-network metric for filter localization, which is physically-meaningful for the flood over a network problem. Using this metric, we further evaluate the simultaneous estimation of channel friction and spatially-variable channel bathymetry, for which the filter seems able to converge simultaneously to sensible values. Results also indicate that friction is a second order effect in flood inundation models applied to gradually varied flow in large rivers. The study is not conclusive regarding whether in an operational situation the simultaneous estimation of friction and bathymetry helps the current forecast. Overall, the results indicate the feasibility of stand-alone EO-based operational flood forecasting.

Comparing hybrid data assimilation methods on the Lorenz 1963 model with increasing nonlinearity

We systematically compare the performance of ETKF-4DVAR, 4DVAR-BEN and 4DENVAR with respect to two traditional methods (4DVAR and ETKF) and an ensemble transform Kalman smoother (ETKS) on the Lorenz 1963 model. We specifically investigated this performance with increasing nonlinearity and using a quasi-static variational assimilation algorithm as a comparison. Using the analysis root mean square error (RMSE) as a metric, these methods have been compared considering (1) assimilation window length and observation interval size and (2) ensemble size to investigate the influence of hybrid background error covariance matrices and nonlinearity on the performance of the methods. For short assimilation windows with close to linear dynamics, it has been shown that all hybrid methods show an improvement in RMSE compared to the traditional methods. For long assimilation window lengths in which nonlinear dynamics are substantial, the variational framework can have diffculties fnding the global minimum of the cost function, so we explore a quasi-static variational assimilation (QSVA) framework. Of the hybrid methods, it is seen that under certain parameters, hybrid methods which do not use a climatological background error covariance do not need QSVA to perform accurately. Generally, results show that the ETKS and hybrid methods that do not use a climatological background error covariance matrix with QSVA outperform all other methods due to the full flow dependency of the background error covariance matrix which also allows for the most nonlinearity.

The effect of the equivalent-weights particle filter on dynamical balance in a primitive equation model

The disadvantage of the majority of data assimilation schemes is the assumption that the conditional probability density function of the state of the system given the observations [posterior probability density function (PDF)] is distributed either locally or globally as a Gaussian. The advantage, however, is that through various different mechanisms they ensure initial conditions that are predominantly in linear balance and therefore spurious gravity wave generation is suppressed. The equivalent-weights particle filter is a data assimilation scheme that allows for a representation of a potentially multimodal posterior PDF. It does this via proposal densities that lead to extra terms being added to the model equations and means the advantage of the traditional data assimilation schemes, in generating predominantly balanced initial conditions, is no longer guaranteed. This paper looks in detail at the impact the equivalent-weights particle filter has on dynamical balance and gravity wave generation in a primitive equation model. The primary conclusions are that (i) provided the model error covariance matrix imposes geostrophic balance, then each additional term required by the equivalent-weights particle filter is also geostrophically balanced; (ii) the relaxation term required to ensure the particles are in the locality of the observations has little effect on gravity waves and actually induces a reduction in gravity wave energy if sufficiently large; and (iii) the equivalent-weights term, which leads to the particles having equivalent significance in the posterior PDF, produces a change in gravity wave energy comparable to the stochastic model error. Thus, the scheme does not produce significant spurious gravity wave energy and so has potential for application in real high-dimensional geophysical applications.

Theoretical insight into diagnosing observation error correlations using observation-minus-background and observation-minus-analysis statistics

To improve the quantity and impact of observations used in data assimilation it is necessary to take into account the full, potentially correlated, observation error statistics. A number of methods for estimating correlated observation errors exist, but a popular method is a diagnostic that makes use of statistical averages of observation-minus-background and observation-minus-analysis residuals. The accuracy of the results it yields is unknown as the diagnostic is sensitive to the difference between the exact background and exact observation error covariances and those that are chosen for use within the assimilation. It has often been stated in the literature that the results using this diagnostic are only valid when the background and observation error correlation length scales are well separated. Here we develop new theory relating to the diagnostic. For observations on a 1D periodic domain we are able to the show the effect of changes in the assumed error statistics used in the assimilation on the estimated observation error covariance matrix. We also provide bounds for the estimated observation error variance and eigenvalues of the estimated observation error correlation matrix. We demonstrate that it is still possible to obtain useful results from the diagnostic when the background and observation error length scales are similar. In general, our results suggest that when correlated observation errors are treated as uncorrelated in the assimilation, the diagnostic will underestimate the correlation length scale. We support our theoretical results with simple illustrative examples. These results have potential use for interpreting the derived covariances estimated using an operational system.

Diagnosing observation error correlations for Doppler radar radial winds in the Met Office UKV model using observation-minus-background and observation-minus-analysis statistics

With the development of convection-permitting numerical weather prediction the efficient use of high resolution observations in data assimilation is becoming increasingly important. The operational assimilation of these observations, such as Dopplerradar radial winds, is now common, though to avoid violating the assumption of un- correlated observation errors the observation density is severely reduced. To improve the quantity of observations used and the impact that they have on the forecast will require the introduction of the full, potentially correlated, error statistics. In this work, observation error statistics are calculated for the Doppler radar radial winds that are assimilated into the Met Office high resolution UK model using a diagnostic that makes use of statistical averages of observation-minus-background and observation-minus-analysis residuals. This is the first in-depth study using the diagnostic to estimate both horizontal and along-beam correlated observation errors. By considering the new results obtained it is found that the Doppler radar radial wind error standard deviations are similar to those used operationally and increase as the observation height increases. Surprisingly the estimated observation error correlation length scales are longer than the operational thinning distance. They are dependent on both the height of the observation and on the distance of the observation away from the radar. Further tests show that the long correlations cannot be attributed to the use of superobservations or the background error covariance matrix used in the assimilation. The large horizontal correlation length scales are, however, in part, a result of using a simplified observation operator.

Sparse density estimation on multinomial manifold combining local component analysis

A new sparse kernel density estimator is introduced based on the minimum integrated square error criterion combining local component analysis for the finite mixture model. We start with a Parzen window estimator which has the Gaussian kernels with a common covariance matrix, the local component analysis is initially applied to find the covariance matrix using expectation maximization algorithm. Since the constraint on the mixing coefficients of a finite mixture model is on the multinomial manifold, we then use the well-known Riemannian trust-region algorithm to find the set of sparse mixing coefficients. The first and second order Riemannian geometry of the multinomial manifold are utilized in the Riemannian trust-region algorithm. Numerical examples are employed to demonstrate that the proposed approach is effective in constructing sparse kernel density estimators with competitive accuracy to existing kernel density estimators.

The Evolution of Modularity in the Mammalian Skull I: Morphological Integration Patterns and Magnitudes

Morphological integration refers to the modular structuring of inter-trait relationships in an organism, which could bias the direction and rate of morphological change, either constraining or facilitating evolution along certain dimensions of the morphospace. Therefore, the description of patterns and magnitudes of morphological integration and the analysis of their evolutionary consequences are central to understand the evolution of complex traits. Here we analyze morphological integration in the skull of several mammalian orders, addressing the following questions: are there common patterns of inter-trait relationships? Are these patterns compatible with hypotheses based on shared development and function? Do morphological integration patterns and magnitudes vary in the same way across groups? We digitized more than 3,500 specimens spanning 15 mammalian orders, estimated the correspondent pooled within-group correlation and variance/covariance matrices for 35 skull traits and compared those matrices among the orders. We also compared observed patterns of integration to theoretical expectations based on common development and function. Our results point to a largely shared pattern of inter-trait correlations, implying that mammalian skull diversity has been produced upon a common covariance structure that remained similar for at least 65 million years. Comparisons with a rodent genetic variance/covariance matrix suggest that this broad similarity extends also to the genetic factors underlying phenotypic variation. In contrast to the relative constancy of inter-trait correlation/covariance patterns, magnitudes varied markedly across groups. Several morphological modules hypothesized from shared development and function were detected in the mammalian taxa studied. Our data provide evidence that mammalian skull evolution can be viewed as a history of inter-module parcellation, with the modules themselves being more clearly marked in those lineages with lower overall magnitude of integration. The implication of these findings is that the main evolutionary trend in the mammalian skull was one of decreasing the constraints to evolution by promoting a more modular architecture.

Improved testing inference in mixed linear models

Mixed linear models are commonly used in repeated measures studies. They account for the dependence amongst observations obtained from the same experimental unit. Often, the number of observations is small, and it is thus important to use inference strategies that incorporate small sample corrections. In this paper, we develop modified versions of the likelihood ratio test for fixed effects inference in mixed linear models. In particular, we derive a Bartlett correction to such a test, and also to a test obtained from a modified profile likelihood function. Our results generalize those in [Zucker, D.M., Lieberman, O., Manor, O., 2000. Improved small sample inference in the mixed linear model: Bartlett correction and adjusted likelihood. Journal of the Royal Statistical Society B, 62,827-838] by allowing the parameter of interest to be vector-valued. Additionally, our Bartlett corrections allow for random effects nonlinear covariance matrix structure. We report simulation results which show that the proposed tests display superior finite sample behavior relative to the standard likelihood ratio test. An application is also presented and discussed. (C) 2008 Elsevier B.V. All rights reserved.

Bias correction in a multivariate normal regression model with general parameterization

This paper derives the second-order biases Of maximum likelihood estimates from a multivariate normal model where the mean vector and the covariance matrix have parameters in common. We show that the second order bias can always be obtained by means of ordinary weighted least-squares regressions. We conduct simulation studies which indicate that the bias correction scheme yields nearly unbiased estimators. (C) 2009 Elsevier B.V. All rights reserved.

Uso da análise discriminante regularizada (RDA) no reconhecimento de padrões em imagens digitais hiperespectral de sensoriamento remoto

Em cenas naturais, ocorrem com certa freqüência classes espectralmente muito similares, isto é, os vetores média são muito próximos. Em situações como esta, dados de baixa dimensionalidade (LandSat-TM, Spot) não permitem uma classificação acurada da cena. Por outro lado, sabe-se que dados em alta dimensionalidade [FUK 90] tornam possível a separação destas classes, desde que as matrizes covariância sejam suficientemente distintas. Neste caso, o problema de natureza prática que surge é o da estimação dos parâmetros que caracterizam a distribuição de cada classe. Na medida em que a dimensionalidade dos dados cresce, aumenta o número de parâmetros a serem estimados, especialmente na matriz covariância. Contudo, é sabido que, no mundo real, a quantidade de amostras de treinamento disponíveis, é freqüentemente muito limitada, ocasionando problemas na estimação dos parâmetros necessários ao classificador, degradando portanto a acurácia do processo de classificação, na medida em que a dimensionalidade dos dados aumenta. O Efeito de Hughes, como é chamado este fenômeno, já é bem conhecido no meio científico, e estudos vêm sendo realizados com o objetivo de mitigar este efeito. Entre as alternativas propostas com a finalidade de mitigar o Efeito de Hughes, encontram-se as técnicas de regularização da matriz covariância. Deste modo, técnicas de regularização para a estimação da matriz covariância das classes, tornam-se um tópico interessante de estudo, bem como o comportamento destas técnicas em ambientes de dados de imagens digitais de alta dimensionalidade em sensoriamento remoto, como por exemplo, os dados fornecidos pelo sensor AVIRIS. Neste estudo, é feita uma contextualização em sensoriamento remoto, descrito o sistema sensor AVIRIS, os princípios da análise discriminante linear (LDA), quadrática (QDA) e regularizada (RDA) são apresentados, bem como os experimentos práticos dos métodos, usando dados reais do sensor. Os resultados mostram que, com um número limitado de amostras de treinamento, as técnicas de regularização da matriz covariância foram eficientes em reduzir o Efeito de Hughes. Quanto à acurácia, em alguns casos o modelo quadrático continua sendo o melhor, apesar do Efeito de Hughes, e em outros casos o método de regularização é superior, além de suavizar este efeito. Esta dissertação está organizada da seguinte maneira: No primeiro capítulo é feita uma introdução aos temas: sensoriamento remoto (radiação eletromagnética, espectro eletromagnético, bandas espectrais, assinatura espectral), são também descritos os conceitos, funcionamento do sensor hiperespectral AVIRIS, e os conceitos básicos de reconhecimento de padrões e da abordagem estatística. No segundo capítulo, é feita uma revisão bibliográfica sobre os problemas associados à dimensionalidade dos dados, à descrição das técnicas paramétricas citadas anteriormente, aos métodos de QDA, LDA e RDA, e testes realizados com outros tipos de dados e seus resultados.O terceiro capítulo versa sobre a metodologia que será utilizada nos dados hiperespectrais disponíveis. O quarto capítulo apresenta os testes e experimentos da Análise Discriminante Regularizada (RDA) em imagens hiperespectrais obtidos pelo sensor AVIRIS. No quinto capítulo são apresentados as conclusões e análise final. A contribuição científica deste estudo, relaciona-se à utilização de métodos de regularização da matriz covariância, originalmente propostos por Friedman [FRI 89] para classificação de dados em alta dimensionalidade (dados sintéticos, dados de enologia), para o caso especifico de dados de sensoriamento remoto em alta dimensionalidade (imagens hiperespectrais). A conclusão principal desta dissertação é que o método RDA é útil no processo de classificação de imagens com dados em alta dimensionalidade e classes com características espectrais muito próximas.