Wavelet analysis on the circle

International audienceThe construction of a wavelet analysis over the circle is presented. The spaces of infinitely times differentiable functions, tempered distributions, and square integrable functions over the circle are analyzed by means of the wavelet transform

Wavelets on Discrete Fields

International audienceAn arithmetic version of continuous wavelet analysis is described. Starting from a square-integrable representation of the affine group of Z Z p (or Z Z) it is shown how wavelet decom-positions of ℓ 2 (Z Z p) can be obtained. Moreover, a redefinition of the dilation operator on ℓ 2 (Z Z p) directly yields an algorithmic structure similar to that appearing with multiresolution analyses

Robust detection and verification of linear relationships to generate metabolic networks using estimates of technical errors

<p>Abstract</p> <p>Background</p> <p>The size and magnitude of the metabolome, the ratio between individual metabolites and the response of metabolic networks is controlled by multiple cellular factors. A tight control over metabolite ratios will be reflected by a linear relationship of pairs of metabolite due to the flexibility of metabolic pathways. Hence, unbiased detection and validation of linear metabolic variance can be interpreted in terms of biological control. For robust analyses, criteria for rejecting or accepting linearities need to be developed despite technical measurement errors. The entirety of all pair wise linear metabolic relationships then yields insights into the network of cellular regulation.</p> <p>Results</p> <p>The Bayesian law was applied for detecting linearities that are validated by explaining the residues by the degree of technical measurement errors. Test statistics were developed and the algorithm was tested on simulated data using 3–150 samples and 0–100% technical error. Under the null hypothesis of the existence of a linear relationship, type I errors remained below 5% for data sets consisting of more than four samples, whereas the type II error rate quickly raised with increasing technical errors. Conversely, a filter was developed to balance the error rates in the opposite direction. A minimum of 20 biological replicates is recommended if technical errors remain below 20% relative standard deviation and if thresholds for false error rates are acceptable at less than 5%. The algorithm was proven to be robust against outliers, unlike Pearson's correlations.</p> <p>Conclusion</p> <p>The algorithm facilitates finding linear relationships in complex datasets, which is radically different from estimating linearity parameters from given linear relationships. Without filter, it provides high sensitivity and fair specificity. If the filter is activated, high specificity but only fair sensitivity is yielded. Total error rates are more favorable with deactivated filters, and hence, metabolomic networks should be generated without the filter. In addition, Bayesian likelihoods facilitate the detection of multiple linear dependencies between two variables. This property of the algorithm enables its use as a discovery tool and to generate novel hypotheses of the existence of otherwise hidden biological factors.</p

Correlation dimension of self-similar surfaces and application to Kirchhoff integrals

Abstract For surfaces generated by a class of asymptotically self-similar processes we define a probability measure, supported by the surface. We show that the correlation dimension of that surface measure is linked to the self-similarity exponent almost surely. This result is applied to the Kirchhoff integral well known in scattering from rough surfaces. We show that a certain average of the scattered intensity exhibits almost surely a scaling that allows us to recover the self-similarity index of the surface in an experiment involving only one sample of the surface

Microsaccade characterization using the continuous wavelet transform and principal component analysis

During visual fixation on a target, humans perform miniature (or fixational) eye movements consisting of three components, i.e., tremor, drift, and microsaccades. Microsaccades are high velocity components with small amplitudes within fixational eye movements. However, microsaccade shapes and statistical properties vary between individual observers. Here we show that microsaccades can be formally represented with two significant shapes which we identfied using the mathematical definition of singularities for the detection of the former in real data with the continuous wavelet transform. For character-ization and model selection, we carried out a principal component analysis, which identified a step shape with an overshoot as first and a bump which regulates the overshoot as second component. We conclude that microsaccades are singular events with an overshoot component which can be detected by the continuous wavelet transform