Can wavelets improve the representation of forecast error covariances in variational data assimilation?

Two wavelet-based control variable transform schemes are described and are used to model some important features of forecast error statistics for use in variational data assimilation. The first is a conventional wavelet scheme and the other is an approximation of it. Their ability to capture the position and scale-dependent aspects of covariance structures is tested in a two-dimensional latitude-height context. This is done by comparing the covariance structures implied by the wavelet schemes with those found from the explicit forecast error covariance matrix, and with a non-wavelet- based covariance scheme used currently in an operational assimilation scheme. Qualitatively, the wavelet-based schemes show potential at modeling forecast error statistics well without giving preference to either position or scale-dependent aspects. The degree of spectral representation can be controlled by changing the number of spectral bands in the schemes, and the least number of bands that achieves adequate results is found for the model domain used. Evidence is found of a trade-off between the localization of features in positional and spectral spaces when the number of bands is changed. By examining implied covariance diagnostics, the wavelet-based schemes are found, on the whole, to give results that are closer to diagnostics found from the explicit matrix than from the nonwavelet scheme. Even though the nature of the covariances has the right qualities in spectral space, variances are found to be too low at some wavenumbers and vertical correlation length scales are found to be too long at most scales. The wavelet schemes are found to be good at resolving variations in position and scale-dependent horizontal length scales, although the length scales reproduced are usually too short. The second of the wavelet-based schemes is often found to be better than the first in some important respects, but, unlike the first, it has no exact inverse transform.

Wavelets and the generalization of the variogram

The experimental variogram computed in the usual way by the method of moments and the Haar wavelet transform are similar in that they filter data and yield informative summaries that may be interpreted. The variogram filters out constant values; wavelets can filter variation at several spatial scales and thereby provide a richer repertoire for analysis and demand no assumptions other than that of finite variance. This paper compares the two functions, identifying that part of the Haar wavelet transform that gives it its advantages. It goes on to show that the generalized variogram of order k=1, 2, and 3 filters linear, quadratic, and cubic polynomials from the data, respectively, which correspond with more complex wavelets in Daubechies's family. The additional filter coefficients of the latter can reveal features of the data that are not evident in its usual form. Three examples in which data recorded at regular intervals on transects are analyzed illustrate the extended form of the variogram. The apparent periodicity of gilgais in Australia seems to be accentuated as filter coefficients are added, but otherwise the analysis provides no new insight. Analysis of hyerpsectral data with a strong linear trend showed that the wavelet-based variograms filtered it out. Adding filter coefficients in the analysis of the topsoil across the Jurassic scarplands of England changed the upper bound of the variogram; it then resembled the within-class variogram computed by the method of moments. To elucidate these results, we simulated several series of data to represent a random process with values fluctuating about a mean, data with long-range linear trend, data with local trend, and data with stepped transitions. The results suggest that the wavelet variogram can filter out the effects of long-range trend, but not local trend, and of transitions from one class to another, as across boundaries.

Unsupervised segmentation using Gabor wavelets and statistical features in LIDAR data analysis

In this paper, we address issues in segmentation Of remotely sensed LIDAR (LIght Detection And Ranging) data. The LIDAR data, which were captured by airborne laser scanner, contain 2.5 dimensional (2.5D) terrain surface height information, e.g. houses, vegetation, flat field, river, basin, etc. Our aim in this paper is to segment ground (flat field)from non-ground (houses and high vegetation) in hilly urban areas. By projecting the 2.5D data onto a surface, we obtain a texture map as a grey-level image. Based on the image, Gabor wavelet filters are applied to generate Gabor wavelet features. These features are then grouped into various windows. Among these windows, a combination of their first and second order of statistics is used as a measure to determine the surface properties. The test results have shown that ground areas can successfully be segmented from LIDAR data. Most buildings and high vegetation can be detected. In addition, Gabor wavelet transform can partially remove hill or slope effects in the original data by tuning Gabor parameters.

Face verification using Gabor wavelets and AdaBoost

This paper presents a new face verification algorithm based on Gabor wavelets and AdaBoost. In the algorithm, faces are represented by Gabor wavelet features generated by Gabor wavelet transform. Gabor wavelets with 5 scales and 8 orientations are chosen to form a family of Gabor wavelets. By convolving face images with these 40 Gabor wavelets, the original images are transformed into magnitude response images of Gabor wavelet features. The AdaBoost algorithm selects a small set of significant features from the pool of the Gabor wavelet features. Each feature is the basis for a weak classifier which is trained with face images taken from the XM2VTS database. The feature with the lowest classification error is selected in each iteration of the AdaBoost operation. We also address issues regarding computational costs in feature selection with AdaBoost. A support vector machine (SVM) is trained with examples of 20 features, and the results have shown a low false positive rate and a low classification error rate in face verification.

Facial feature extraction and selection by Gabor wavelets and boosting

In this paper, we present a feature selection approach based on Gabor wavelet feature and boosting for face verification. By convolution with a group of Gabor wavelets, the original images are transformed into vectors of Gabor wavelet features. Then for individual person, a small set of significant features are selected by the boosting algorithm from a large set of Gabor wavelet features. The experiment results have shown that the approach successfully selects meaningful and explainable features for face verification. The experiments also suggest that for the common characteristics such as eyes, noses, mouths may not be as important as some unique characteristic when training set is small. When training set is large, the unique characteristics and the common characteristics are both important.

Comparison of extrasystolic ECG signal classifiers using discrete wavelet transforms

This work compares and contrasts results of classifying time-domain ECG signals with pathological conditions taken from the MITBIH arrhythmia database. Linear discriminant analysis and a multi-layer perceptron were used as classifiers. The neural network was trained by two different methods, namely back-propagation and a genetic algorithm. Converting the time-domain signal into the wavelet domain reduced the dimensionality of the problem at least 10-fold. This was achieved using wavelets from the db6 family as well as using adaptive wavelets generated using two different strategies. The wavelet transforms used in this study were limited to two decomposition levels. A neural network with evolved weights proved to be the best classifier with a maximum of 99.6% accuracy when optimised wavelet-transform ECG data wits presented to its input and 95.9% accuracy when the signals presented to its input were decomposed using db6 wavelets. The linear discriminant analysis achieved a maximum classification accuracy of 95.7% when presented with optimised and 95.5% with db6 wavelet coefficients. It is shown that the much simpler signal representation of a few wavelet coefficients obtained through an optimised discrete wavelet transform facilitates the classification of non-stationary time-variant signals task considerably. In addition, the results indicate that wavelet optimisation may improve the classification ability of a neural network. (c) 2005 Elsevier B.V. All rights reserved.

A continuous wavelet transform and classification method for delirium motoric subtyping

The usefulness of motor subtypes of delirium is unclear due to inconsistency in subtyping methods and a lack of validation with objective measures of activity. The activity of 40 patients was measured over 24 h with a discrete accelerometer-based activity monitor. The continuous wavelet transform (CWT) with various mother wavelets were applied to accelerometry data from three randomly selected patients with DSM-IV delirium that were readily divided into hyperactive, hypoactive, and mixed motor subtypes. A classification tree used the periods of overall movement as measured by the discrete accelerometer-based monitor as determining factors for which to classify these delirious patients. This data used to create the classification tree were based upon the minimum, maximum, standard deviation, and number of coefficient values, generated over a range of scales by the CWT. The classification tree was subsequently used to define the remaining motoric subtypes. The use of a classification system shows how delirium subtypes can be categorized in relation to overall motoric behavior. The classification system was also implemented to successfully define other patient motoric subtypes. Motor subtypes of delirium defined by observed ward behavior differ in electronically measured activity levels.

Linear-wavelet models for non-linear identification applied to a pressure plant

A nonlinear regression structure comprising a wavelet network and a linear term is proposed for system identification. The theoretical foundation of the approach is laid by proving that radial wavelets are orthogonal to linear functions. A constructive procedure for building such models is described and the approach is tested with experimental data.

Linear-wavelet models applied to the identification of a two-link manipulator

This paper shows that a wavelet network and a linear term can be advantageously combined for the purpose of non linear system identification. The theoretical foundation of this approach is laid by proving that radial wavelets are orthogonal to linear functions. A constructive procedure for building such nonlinear regression structures, termed linear-wavelet models, is described. For illustration, sim ulation data are used to identify a model for a two-link robotic manipulator. The results show that the introduction of wavelets does improve the prediction ability of a linear model.

Linear-wavelet models for system identification

A model structure comprising a wavelet network and a linear term is proposed for nonlinear system identification. It is shown that under certain conditions wavelets are orthogonal to linear functions and, as a result, the two parts of the model can be identified separately. The linear-wavelet model is compared to a standard wavelet network using data from a simulated fermentation process. The results show that the linear-wavelet model yields a smaller modelling error when compared to a wavelet network using the same number of regressors.