Comparison of fitting in L-domain and t-domain of simulated noisy exponentials defined by parameters and , superimposed Poisson noise and stationary gaussian noise with different standard deviations .

<p> is the probability that the fitting parameters obtained in the L-domain approximate the true values better than those obtained in the t-domain (i.e., ) under the indicated noise condition. With optimal importance-weighting for Poisson noise (), the comparison is carried out for four different levels of gaussian noise, given as a ().</p

Double exponential decay analysis in time- and Legendre-domain.

<p>(<b>a,b</b>) The interaction of Ru(II) complexes with DNA (same as in in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0090500#pone-0090500-g001" target="_blank">Fig. 1</a>) shows double exponential decay(<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0090500#pone.0090500-Bazzicalupi1" target="_blank">[12]</a>, data courtesy of F. Secco, Univ Pisa,I). Same data fitted in t- (<b>a</b>) and L-domain (<b>b</b>). (<b>c</b>) Probability for the fitting error in the L-domain, , to be smaller than that in the t-domain, , represented as a function of and . Color code of the probability is shown beneath the plot. For each pair () 1000 trials were computed. (<b>d</b>) Relative difference of fitting errors, , as a function of and . Color code beneath the plot. 1000 trials per pixel. Left of the solid and dashed white lines in <b>a</b> and <b>b</b>, the success rate of the fit in the L-domain is larger that and , respectively. Left of the solid and dashed black lines in <b>c</b> and <b>d</b>, the success rate of the fit in the t-domain is larger that 50% and 95%, respectively. Relative error differences were calculated only for successful trials. (<b>e</b>) Success rate of fitting in the L-domain (solid) and t-domain (dashed) along the vertical line in <b>c</b>, i.e., as a function of with kept at .</p

- dependence of the required number of Legendre components.

<p>(<b>a</b>) Legendre spectra of noisy exponentials with (gray) and (red). Shown are the average Legendre amplitudes obtained from trials. Error bars, standard deviation of the respective component. (<b>b</b>) Coefficient of variation of the components for the two spectra shown in <b>a</b>.</p

Legendre lowpass-filtered EPSP.

<p>(<b>a</b>) Noisy EPSP and its mean simulated as , with , and . (<b>b, d</b>) Legendre spectra of the noisy EPSP (<b>b</b>), and its mean (<b>d</b>). (<b>c</b>) Inverse transform of through of <b>b</b> approximating the EPSP's mean (dashed, red). Gray curve in <b>c</b>, Fourier lowpass () of the noisy EPSP.</p

Comparison of fitting in time- and Legendre - domain.

<p>(<b>a</b>) LMA fit (dashed, red) and mean (cont.) of a noisy exponential with offset in the time domain. Poisson noise; offset, 100; gaussian offset noise with . (<b>b</b>) Legendre spectrum of the same noisy exponential (gray) and LMA fit in the Legendre space (red). (<b>c,d</b>) Probability density function giving the frequency with which the fitted amplitude, normalized to the true value (<b>c</b>), or with which the fitted time constant, normalized to the true value (<b>d</b>)occurs in 5000 trials. True values assumed in c,d: A = 3000, tau = 0.1. All weights set to 1. The red and gray curves give the frequencies resulting from the fits carried out in the time (gray) or Legendre (red) domain, respectively. (<b>e,f</b>) Probability density functions as in (<b>c,d</b>), except that the data were weighted with . The four calculated pdf's in <b>c</b> through <b>f</b> are fitted by Gaussians.</p

Legendre filter of the sum of two exponentials.

<p>(<b>a</b>) Noisy double exponential and its mean (). (<b>b</b>, <b>d</b>) Legendre spectra of the noisy double exponential (<b>b</b>) and its mean (<b>d</b>). (<b>c</b>) Mean of <b>a</b> (cont.) and lowpass-filtered Legendre spectrum in <b>b</b> (red, dashed).</p

Filtering exponentials and Legendre lowpass.

<p>(<b>a</b>) Double exponential decay during a stopped-flow recording. The reaction monitored is the interaction of ruthenium complexes with DNA, scaled to the interval [−1, 1]. For the experimental details of the system, see [12]. (<b>b</b>) The first 17 components of the Legendre spectrum of . The inverse fLT of the components through of the spectrum (<b>b</b>) gives the red curve in (<b>a</b>). Note that the sharp peak in the noisy trace is virtually not reflected in the filtered curve. (<b>c</b>) Autocorrelation curve (gray) resulting from an experiment where the diffusion constant of tetramethylrhodamine was measured (own data). In this example, fLT and ifLT are performed for non-equidistant samples, and we re-scaled the x-axis to correlation delays. (<b>d</b>) Legendre spectrum of the ACF shown in (<b>c</b>). The red curve in (<b>c</b>) is the inverse fLT of the components through of the Legendre spectrum.</p

Filtering exponentials convolved with a system response function.

<p>(<b>a</b>) Exponential on the interval [−1, 1] with Poisson noise added. Amplitude, , time constant . (<b>b</b>) Legendre spectrum of x as resulting from <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0090500#pone.0090500.e283" target="_blank">eq. 5</a>. (<b>c</b>) Mean of (continuous) and inverse fLT (dashed, <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0090500#pone.0090500.e306" target="_blank">eq. 6</a>) of through of the spectrum shown in <b>b</b>. (<b>d</b>) Legendre spectrum of the mean , largely lacking higher noise components. (<b>e</b>) Noisy curve is the convolution of with and . was chosen such that the curve overlaps with for large . (<b>f</b>) Legendre spectrum (gray bars) of convoluted noisy exponential shown in <b>e</b> (continuous curve). The lowpass-filtered inverse transform is shown in <b>e</b> (continuous curve) and approximates the convoluted noisy exponential. In addition, <b>f</b> shows the Legendre spectrum of , obtained through <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0090500#pone.0090500.e321" target="_blank">eq. 9</a>. The lowpass-filtered inverse transform of this spectrum is shown as the red dashed curve in <b>e</b> and approximates the original non-convoluted exponential, from which the noisy convoluted curve was generated.</p

Computational cost of fitting in time - and Legendre - domain.

<p>Dots and asterisks indicate the computational cost in Legendre - and time - domain, respectively, on a logarithmic scale. Black and red curves refer to whether (red) or not (black) a system response function was taken into account. Each point or asterisk is the average duration of computations. As the analysis in the time - domain is fastest when using the Fast Fourier Transform, we choose the sample sizes to be powers of , starting with . Inset, left, FLIM image of a mouse hippocampus neuron probed for oxidative stress using the hydrogen peroxide sensor HyPer <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0090500#pone.0090500-Belousov1" target="_blank">[13]</a> and, right, fluorescence lifetime function for one pixel in the middle of the cell (150 samples, data courtesy K. Kizina and M. Müller, CNMPB, Göttingen).</p

Distribution of diffusion coefficients.

<p>Blue: histogram of best-fit results for diffusion coefficients of fluorescein from 35 cilia, with mean and standard deviation as dark gray line and light gray area, respectively. The diffusion coefficient of fluorescein in aqueous solution at 25°C <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0039628#pone.0039628-Culbertson1" target="_blank">[14]</a> is shown in red, while values corrected for a range of (23±1)°C are shown in green.</p