Cumulant | EPFL Graph Search

In probability theory and statistics, the cumulants κn of a probability distribution are a set of quantities that provide an alternative to the moments of the distribution. Any two probability distributions whose moments are identical will have identical cumulants as well, and vice versa. The first cumulant is the mean, the second cumulant is the variance, and the third cumulant is the same as the third central moment. But fourth and higher-order cumulants are not equal to central moments. In some cases theoretical treatments of problems in terms of cumulants are simpler than those using moments. In particular, when two or more random variables are statistically independent, the n-th-order cumulant of their sum is equal to the sum of their n-th-order cumulants. As well, the third and higher-order cumulants of a normal distribution are zero, and it is the only distribution with this property. Just as for moments, where joint moments are used for collections

Balanced Super-resolution Optical Fluctuation Imaging (bSOFI)

Corinne Berclaz, Noelia Laura Bocchio, Claudio Dellagiacoma, Matthias Geissbühler, Stefan Geissbühler, Theo Lasser, Marcel Leutenegger

Super-resolution optical fluctuation imaging (SOFI) achieves 3D super-resolution by computing higher-order cumulants of stochastically blinking fluorophores. In contrast to localization microscopy, SOFI is compatible with weakly emitting fluorophores and a wide range of blinking conditions. The main drawback of SOFI is the nonlinear response to brightness and blinking heterogeneities in the sample, which limits the use of higher cumulant orders. Balanced super-resolution optical fluctuation imaging (bSOFI), extends SOFI by the combination of several cumulant orders to map fluorescence-related molecular statistics, such as molecular state lifetimes, concentrations and brightness distributions with super-resolution. Since these parameters are often linked to the chemical microenvironment of the fluorophores, they report on static differences and/or dynamic changes within cells and thus add a “functional” dimension to super-resolution microscopy based on stochastic switching. Furthermore, the information obtained can be used to correct for the nonlinear brightness and blinking response of cumulants. We show experimental results of Alexa647-labeled microtubules in fixed HeLa cells with an up to five-fold resolution improvement compared to diffraction-limited widefield microscopy. Using a total-internal-reflection illumination scheme, we obtain depth information through the estimation of the spatial distributions of the molecular brightness as well as the blinking on-ratio.

2012

Mapping molecular statistics with balanced super-resolution optical fluctuation imaging (bSOFI)

Corinne Berclaz, Noelia Laura Bocchio, Claudio Dellagiacoma, Stefan Geissbuehler, Theo Lasser, Marcel Leutenegger

Super-resolution optical fluctuation imaging (SOFI) achieves 3D super-resolution by computing temporal cumulants or spatio-temporal cross-cumulants of stochastically blinking fluorophores. In contrast to localization microscopy, SOFI is compatible with weakly emitting fluorophores and a wide range of blinking conditions. The main drawback of SOFI is the nonlinear response to brightness and blinking heterogeneities in the sample, which limits the use of higher cumulant orders for improving the resolution. Balanced super-resolution optical fluctuation imaging (bSOFI) analyses several cumulant orders for extracting molecular parameter maps, such as molecular state lifetimes, concentration and brightness distributions of fluorophores within biological samples. Moreover, the estimated blinking statistics are used to balance the image contrast, i.e. linearize the brightness and blinking response and to obtain a resolution improving linearly with the cumulant order. Using a widefield total-internal-reflection (TIR) fluorescence microscope, we acquired image sequences of fluorescently labelled microtubules in fixed HeLa cells. We demonstrate an up to five-fold resolution improvement as compared to the diffraction-limited image, despite low single-frame signal-to-noise ratios. Due to the TIR illumination, the intensity profile in the sample decreases exponentially along the optical axis, which is reported by the estimated spatial distributions of the molecular brightness as well as the blinking on-ratio. Therefore, TIR-bSOFI also encodes depth information through these parameter maps. bSOFI is an extended version of SOFI that cancels the nonlinear response to brightness and blinking heterogeneities. The obtained balanced image contrast significantly enhances the visual perception of super-resolution based on higher-order cumulants and thereby facilitates the access to higher resolutions. Furthermore, bSOFI provides microenvironment-related molecular parameter maps and paves the way for functional super-resolution microscopy based on stochastic switching.

2012

A characterization of the normal distribution using stationary max-stable processes

Sebastian Engelke

Consider a max-stable process of the form , , where are points of the Poisson process with intensity u (-2)du on (0,a), X (i) , , are independent copies of a random d-variate vector X (that are independent of the Poisson process), and is a function. We show that the process eta is stationary if and only if X has multivariate normal distribution and kappa(t)-kappa(0) is the cumulant generating function of X. In this case, eta is a max-stable process introduced by R. L. Smith.

Springer Verlag2016

Mapping molecular statistics with balanced super-resolution optical fluctuation imaging (bSOFI)

Corinne Berclaz, Noelia Laura Bocchio, Claudio Dellagiacoma, Stefan Geissbuehler, Theo Lasser, Marcel Leutenegger

2012