Hi Jakub,
- The first plot (Ac_{fin,sys} is a tail-corrected autocorrelation function. Autocorrelation tails can be noisy and for longer time-scales move below and above the x-axis as shown in the upper image here
The first column shows an attempt to correct the autocorrelation function tail behaviour, the second column shows the raw data for the autocorrelation function. The third column shows the correlation between contacts and hydrogen bonds, telling you how many contacts lead to a hydrogen bond in your two analysed groups. The fourth plot is a simple numerical derivative of the first column.
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In your specific case, the negative values here are due to the tail correction, but generally autocorrelation functions exhibit the behaviour shown above, where for long time-scales curves decay in a sinoidal pattern, showing also negative values.
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The second column is the “true” autocorrelation function, the first one the corrected one Erik Marklund thought you should use to analyse time-scales. Fitting autocorrelation functions with noisy tails is a science in itself, for in-depth reading have a look here.
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To determine life-times, you would like to match an exponential function where you can read off the life-time from the exponent \tau from \exp^{-\frac{t}{\tau}} to the autocorrelation curve that you obtained.
One approach for stable fitting exponential decays with single exponentials is to match the areas of an ideal to the area under your autocorrelation function, cutting off noise tails or correcting for them. Some other ideas for calculating the life-time you can have a look here: