Figure 2.
Noise broadens regularized rate spectra. (top) Regularized rate spectra (using ridge regression) for the tri-exponential data with increasing amounts of noise N(0, s2), for s = 0.001, 0.05 and 0.2. (middle) Rate spectra shown are expectation values from posterior sampling, shown with a 95% confidence interval (in all cases very small). Despite this spectral broadening, regularized rate spectra show three peaks corresponding to each timescale, even for very noisy data. (bottom) The posterior distributions sampled for σ and τ give additional information about the nature of the data being fit. While the posterior distribution of τ remains robust for different values of synthetic noise (reflecting the regularization penalty), the posterior distributions of σ are very narrow and predict extremely well the variance of the artificial noise used to product the time series.