Autocorrelation

The numpy.autocorrelate() only works with one-dimensional time series. Sometimes we are interested in computing autocorrelation for time-series of multidimensional vectors. We can do that by considering each component independently and summing the autocorrelation over all components.

This pipelines is designed to compute autocorrelation functions for time-dependent vectors. Make sure your vectors are normalized since vectors can be correlated in a sense of their direction. In addition to vector_autocorrelate(ts_array), the code also contains the function index_autocorrelate(index_ts) which acts on the time-series of the indexes, for example, an atom or a molecule that visits a specific volume. 0 means nothing is present in the volume. The code can compute autocorrelation by one-hot-encoding all indexes and applying a simple vector correlation function. The average lifetime of the same index (vector direction) can be estimated from the resulting autocorrelation by fitting it with exponential exp(-t/tau). tau is the average residence (life) time (of the same index in the volume).

Usage:

Autocorrelation for time-series of normalized vectors:

ts_array = np.array([
            [1.0, 0.0, 0.0], [1.0, 0.0, 0.0], [0.8, 0.6, 0.0], [0.0, 1.0, 0.0]
])
ac = vector_autocorrelate(ts_array)
print(ac) # [1, 0.8, 0.4, 0.0]

Autocorrelation for time-series of the indexes (0 means the molecules is absent from the volume):

index_ts = [35, 35, 36, 0]
ac = index_autocorrelate(index_ts)
print(ac) # [3/4, 1/3, 0, 0]

The resulting ac shows the value of correlation of the signal with itself at different lagtime from 0 to t, if there are t time points in the array. For 3-dimensional vectors, ac[0]==1 since the dot product of the normalized vector with itself equals to 1. However, for the case of indeces, ac[0] shows the occupancy (or, probability to find any index in the volume).

Added new features - estimating autocorrelation times

• Using an exponential fit of the normalized AC function (autocorrelation_time_fit())
• Using an estimator based on AC series summation (autocorrelation_time_sum()) adapted from (https://emcee.readthedocs.io/en/stable/tutorials/autocorr/#autocorr)

Reading

https://chem.libretexts.org/Bookshelves/Biological_Chemistry/Concepts_in_Biophysical_Chemistry_(Tokmakoff)/06%3A_Dynamics_and_Kinetics/22%3A_Biophysical_Reaction_Dynamics/22.05%3A_Time-Correlation_Functions