ner_tags: O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation O O O O O O O O O O B-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation O O O O O O O O O O O O O O B-Model I-Model I-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O
tokens: a function of time (symbols). We use Bayesian analysis (Sect. 2.4 ) to calculate how our oscillation model describes the observational data. Our Bayesian analysis returns the posterior PDF for each of the model parameters. These may be plotted as histograms, such as those for and in Fig. 6 , or summarised by quoting the median and 95% credible intervals as in Table 2 . The red shaded region represents the 99% credible interval for the loop position described by our model, including the estimated noise . The parameters (e.g., vertical dotted line denoting ) or model components (e.g., background trend shown by blue line in top left panel) plotted in figures correspond to the median values of the relevant parameters. For comparison, the results for analysing Loop #1 with a model including the fundamental kink mode only is shown in Fig. 16 . The top right panel shows the wavelet analysis of the signal. The colour contours represent the spectral amplitude (square root of the spectral power). The dashed lines show the time-dependent periods of oscillation , which demonstrate a gradual increase during the course of the oscillation of approximately 20%. The middle right panel shows a 2D histogram of the seismologically determined transverse density profile parameters based on our MCMC model samplings (with intensity normalised to unity). The red bars correspond to the median values and 95% credible intervals based on the individual 1D histograms (bottom panels) and quoted in Table 2 . They give a convenient summmary of the localisation of the parameters by our model fit, though we note that they do not reflect the strong inverse relationship between and in determing the damping rate (see Fig. 2 ). Consequently, the 2D contour (likely density profile parameters) is significantly more localised than considering the posterior credible intervals independently would suggest. It is necessary to appreciate this inverse relationship when considering the quality of the constraint on density profile indicated by the data. For example, is generally less well constrained for lower density contrasts than for larger density contrasts, as discussed in Sect. 2 . We note that the 95th percentiles quoted in this paper correspond to a confidence interval, in contrast to Paper I where the errors were estimated values of . For example, Paper I gives the density contrast for Loop #1 as , and here (Table 2 ) it is , which corresponds to a that is approximately five times smaller. On the other hand, the for the here is comparable to that in Paper I. The solid curves in the histogram plots are fits to the data using the exponentially modified Gaussian function of the form where is the complementary error function, is a constant determining the amplitude, and are the mean and standard deviation of the Gaussian component, respectively, and is the rate of the exponential component. Equation ( 18 ) is found to describe the histogram profiles well, in particular when the