theory.rst

Theory

pycorrelate.pycorrelate

Cross-correlation of point processes

In fluorescence correlation spectroscopy (FCS) the (normalized) cross-correlation function (CCF) of two continuous signals I₁(t) and I₂(t) is defined as:

$$G(\tau) = \frac{\langle I_1(t)\; I_2(t) \rangle} {\langle I_1(t)\rangle\langle I_2(t) \rangle}$$

The auto-correlation function (ACF) is just a special case where I₁(t) = I₂(t).

In actual experiments, signals are not continuous but come from single-photon detectors that produce a pulse for each photon. These pulses are usually timestamped with ~10ns resolution. The series of photon arrival times is used as input for ACF or CCF computations.

In principle, timestamps can be binned to produce a discrete-time signal. In signal processing, the (non-normalized) cross-correlation of two real discrete-time signals {A_i} and {B_i} is defined as

$$c[k] = \sum_{i=0}^{N} A[i]\ B[i+k].$$

The previous formula is implemented by ucorrelate and numpy.correlate.

Binning timestamps to obtain timetraces would be very inefficient for FCS analysis where time-lags spans may orders of magnitude. It is much more efficient to directly compute the cross-correlation function from timestamps. The popular multi-tau algorithm allows computing the correlation directly from timestamps on a fixed arrangement of quasi-log-spaced bins. More generally, Laurence algorithm (Laurence et al. Optics Letters (2006)) allows computing cross-correlation from timestamps on arbitrary bins of time-lags, with similar performances as the multi-tau. Computing cross-correlation C(τ) from timestamps is fundamentally a counting tasks. Given two timestamps arrays t and u and considering the k-th time-lag bin [τ_k, τ_k + 1), C(k) is the number of pairs where:

τ_k ≤ t_i − u_j < τ_k + 1

for all the possible i and j combinations.

$$C(k) = \frac{n(\{(i,j) \ni t_i < u_i - \Delta\tau_k\})}{\Delta\tau_k}$$

where n({}) is the operator counting the elements in a set, Δτ_k is the duration of the k-th time-lag bin and T is the measurement duration. For FCS we normally want the normalized CCF, that is:

$$G(k) = \frac{n(\{(i,j) \ni t_i < u_i - \Delta\tau_k\})} {n(\{i \ni t_i \le T - \Delta\tau_k\})\:n(\{j \ni u_j \ge \Delta\tau_k\})} \frac{(T-\Delta\tau_k)}{\Delta\tau_k}$$

Eq. Ck and Gk are implemented by pcorrelate, where the argument normalize allows choosing between the normalized and unnormalized version.

Note

In Laurence 2006 the expression for G(k) (there called C_AB(τ)) does not include the Δτ_k in the denominator due to a typo.

References

• Laurence, T. A., Fore, S., Huser, T. (2006). Fast, flexible algorithm for calculating photon correlations. Optics Letters , 31 (6), 829–831. https://doi.org/10.1364/OL.31.000829
• Petra Schwille and Elke Haustein, Fluorescence Correlation Spectroscopy An Introduction to its Concepts and Applications
• Haustein, E., Schwille, P. (2003). Ultrasensitive investigations of biological systems by fluorescence correlation spectroscopy. Methods, 29 (2), 153–166. https://doi.org/10.1016/S1046-2023(02)00306-7