We describe how the true statistical error on an average of correlated data can be obtained with ease and efficiency by a renormalization group method. The method is illustrated with numerical and analytical examples, having finite as well as infinite range correlations.

We optimally localize isolated fluorescent beads and molecules imaged as diffraction-limited spots, determine the orientation of molecules, and present reliable formulae for the precisions of various localization methods. For beads, theory and experimental data both show that unweighted least-squares fitting of a Gaussian squanders one third of the available information, a popular formula for its precision exaggerates beyond Fisher's information limit, and weighted least-squares may do worse, while maximum likelihood fitting is practically optimal.

The force exerted by an optical trap on a dielectric bead in a fluid is often found by fitting a Lorentzian to the power spectrum of Brownian motion of the bead in the trap. We present explicit functions of the experimental power spectrum that give the values of the parameters fitted, including error bars and correlations, for the best such 2 fit in a given frequency range. We use these functions to determine the information content of various parts of the power spectrum, and find, at odds with lore, much information at relatively high frequencies. Applying the method to real data, we obtain perfect fits and calibrate tweezers with less than 1% error when the trapping force is not too strong. Relatively strong traps have power spectra that cannot be fitted properly with any Lorentzian, we find. This underscores the need for better understanding of the power spectrum than the Lorentzian provides. This is achieved using old and new theory for Brownian motion in an incompressible fluid, and new results for a popular photodetection system. The trap and photodetection system are then calibrated simultaneously in a manner that makes optical tweezers a tool of precision for force spectroscopy, local viscometry, and probably other applications.

Supplementary data are available at Bioinformatics online.

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Experimental time series for trajectories of motile cells may contain so much information that a systematic analysis will yield cell-type-specific motility models. Here we demonstrate how, using human keratinocytes and fibroblasts as examples. The two resulting models reflect the cells' different roles in the organism, it seems, and show that a cell has a memory of past velocities. They also suggest how to distinguish quantitatively between various surfaces' compatibility with the two cell types.

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