Ed A. K. Cohen

A Statistical Study of Temporally Smoothed Wavelet Coherence

The use of the wavelet coherence of two series in hypothesis testing relies on some sort of smoothing being carried out in order that the coherence estimator is not simply unity. A previous study considered averaging via the use of multiple Morse wavelets. Here we consider time-domain smoothing and use of a single Morlet wavelet. Since the Morlet wavelet is complex-valued, we derive analytic results for the case of wavelet coherence calculated from complex-valued, jointly stationary and Gaussian time series. The temporally smoothed wavelet coherence can be written in terms of Welchs overlapping segment averaging (WOSA) spectrum estimators, and by using multitaper equivalent representations for the WOSA estimators we show that Goodmans distribution is appropriate asymptotically, and readily derive the appropriate degrees of freedom. The theoretical results are verified via simulations and illustrated using solar physics data.

Method for co-cluster analysis in multichannel single-molecule localisation data

Abstract We demonstrate a combined univariate and bivariate Getis and Franklin’s local point pattern analysis method to investigate the co-clustering of membrane proteins in two-dimensional single-molecule localisation data. This method assesses the degree of clustering of each molecule relative to its own species and relative to a second species. Using simulated data, we show that this approach can quantify the degree of cluster overlap in multichannel point patterns. The method is validated using photo-activated localisation microscopy and direct stochastic optical reconstruction microscopy data of the proteins Lck and CD45 at the T cell immunological synapse. Analysing co-clustering in this manner is generalizable to higher numbers of fluorescent species and to three-dimensional or live cell data sets.

A Statistical Analysis of Morse Wavelet Coherence

Wavelet coherence computed from two time series has been widely applied in hypothesis testing situations, but has proven resistant to analytic study, with resort to simulations for statistical properties. As part of the null hypothesis being tested, such simulations invariably assume joint stationarity of the series. If estimated using multiple orthogonal Morse wavelets, wavelet coherence is in fact amenable to statistical study. Since the wavelets are complex-valued, we consider the case of wavelet coherence calculated from discrete-time complex-valued and stationary time series. Under Gaussianity, the Goodman distribution is, for large samples, appropriate for wavelet coherence. The true wavelet coherence value is identified in terms of its frequency domain equivalent. Theoretical results are illustrated and verified via simulations.

Wavelet Coherence for Certain Nonstationary Bivariate Processes

A previous study considered the estimation of wavelet coherence from jointly stationary time series via time-domain smoothing and use of a single Morlet wavelet. The form of the asymptotic (Goodmans) distribution was derived. In this paper we extend this approach to nonstationary time series where the nonstationarity is induced by various types of modulation. The model forms of coherence studied include constant over time and scale, time-varying, scale-varying, and time-and-scale varying. These coherence models are carefully derived from appropriate statistical models for nonstationary processes. The portion of the signals used in calculating the coherence at a scale a depends on a ; provided its size-or equivalently the number of degrees of freedom of the estimator-is appropriate to the time variation in coherence at that scale, good estimation results are achieved. Moreover, Goodmans distribution is seen still to be appropriate for the estimator.

Analysis of Point Based Image Registration Errors With Applications in Single Molecule Microscopy

We present an asymptotic treatment of errors involved in point-based image registration where control point (CP) localization is subject to heteroscedastic noise; a suitable model for image registration in fluorescence microscopy. Assuming an affine transform, CPs are used to solve a multivariate regression problem. With measurement errors existing for both sets of CPs this is an errors-in-variable problem and linear least squares is inappropriate; the correct method being generalized least squares. To allow for point dependent errors the equivalence of a generalized maximum likelihood and heteroscedastic generalized least squares model is achieved allowing previously published asymptotic results to be extended to image registration. For a particularly useful model of heteroscedastic noise where covariance matrices are scalar multiples of a known matrix (including the case where covariance matrices are multiples of the identity) we provide closed form solutions to estimators and derive their distribution. We consider the target registration error (TRE) and define a new measure called the localization registration error (LRE) believed to be useful, especially in microscopy registration experiments. Assuming Gaussianity of the CP localization errors, it is shown that the asymptotic distribution for the TRE and LRE are themselves Gaussian and the parameterized distributions are derived. Results are successfully applied to registration in single molecule microscopy to derive the key dependence of the TRE and LRE variance on the number of CPs and their associated photon counts. Simulations show asymptotic results are robust for low CP numbers and non-Gaussianity. The method presented here is shown to outperform GLS on real imaging data.

3D Bayesian cluster analysis of super-resolution data reveals LAT recruitment to the T cell synapse

Single-molecule localisation microscopy (SMLM) allows the localisation of fluorophores with a precision of 10–30 nm, revealing the cell’s nanoscale architecture at the molecular level. Recently, SMLM has been extended to 3D, providing a unique insight into cellular machinery. Although cluster analysis techniques have been developed for 2D SMLM data sets, few have been applied to 3D. This lack of quantification tools can be explained by the relative novelty of imaging techniques such as interferometric photo-activated localisation microscopy (iPALM). Also, existing methods that could be extended to 3D SMLM are usually subject to user defined analysis parameters, which remains a major drawback. Here, we present a new open source cluster analysis method for 3D SMLM data, free of user definable parameters, relying on a model-based Bayesian approach which takes full account of the individual localisation precisions in all three dimensions. The accuracy and reliability of the method is validated using simulated data sets. This tool is then deployed on novel experimental data as a proof of concept, illustrating the recruitment of LAT to the T-cell immunological synapse in data acquired by iPALM providing ~10 nm isotropic resolution.

Multi-wavelet coherence for point processes on the real line

Coherence is a well established measure of linear dependency between a pair of stationary random processes in the frequency domain. Wavelet coherence measures the linear dependency between a pair of signals in time-scale space and is therefore more suitable for non-stationary processes. Until now it has only been considered in relation to regularly sampled ordinary time-series. Here, for the first time, it is applied to point processes on the real line. We consider smoothing the individual wavelet spectra by averaging over a set of orthogonal Morse wavelets and show that under the assumption of independent Poisson processes the Goodman distribution is appropriate.

Cramér-Rao Lower Bound for Point Based Image Registration With Heteroscedastic Error Model for Application in Single Molecule Microscopy

The Cramér-Rao lower bound for the estimation of the affine transformation parameters in a multivariate heteroscedastic errors-in-variables model is derived. The model is suitable for feature-based image registration in which both sets of control points are localized with errors whose covariance matrices vary from point to point. With focus given to the registration of fluorescence microscopy images, the Cramér-Rao lower bound for the estimation of a features position (e.g., of a single molecule) in a registered image is also derived. In the particular case where all covariance matrices for the localization errors are scalar multiples of a common positive definite matrix (e.g., the identity matrix), as can be assumed in fluorescence microscopy, then simplified expressions for the Cramér-Rao lower bound are given. Under certain simplifying assumptions these expressions are shown to match asymptotic distributions for a previously presented set of estimators. Theoretical results are verified with simulations and experimental data.

Image registration error analysis with applications in single molecule microscopy

This paper is concerned with assessing localization errors emanating from the image registration of two monochromatic fluorescence microscopy images. Assuming an affine transform exists between images, registration in this setting typically involves using control points to solve a multivariate linear regression problem; however with measurement errors existing in both sets of variables the use of linear least squares is inappropriate. It is shown that image registration is an errors-in-variable problem and as such the correct method is to use generalized least squares. Traditionally this requires the measurement errors to be independent and identically distributed (iid); an assumption that is rarely satisfied in practical situations. An extension of the multivariate generalized least squares estimator that allows non-iid noise is applied. The distributional properties of the estimators are used to derive localization errors emanating from the image registration process in terms of photon counts and experimental parameters.

Statistical Properties for Coherence Estimators From Evolutionary Spectra

Evolutionary spectra were developed by Priestley to extend spectral analysis to some nonstationary time series, in particular semistationary processes, of which the ubiquitous uniformly modulated processes are a subclass. Coherence is well defined for bivariate semistationary processes and can be estimated from such processes. We consider Priestleys estimator for the evolutionary spectral density matrix, and show that its elements can be written as weighted multitaper estimators with calculable weights and tapers. Under Gaussianity an approximating Wishart-distribution model follows for the spectral matrix, valid for all frequencies except small computable intervals near zero and Nyquist. Moreover, the critically important degrees of freedom are known. Consequently, the statistical distribution of the coherence is given by Goodmans distribution and the raw coherence estimate can be accurately debiased. Theoretical results are verified using a model for wind fluctuations: Simulations give excellent agreement between the mean debiased coherence estimates and true coherence, and between the proposed and empirical distributions of coherence.