Autocorrelation uncorrelated data

Uncorrelated Data In the first step of data analysis, it should be checked whether the data are uncorrelated or correlated. Uncorrelated data do not show any trends in their autocorrelation function (Figure 3.23). Note how small the r(r) values are for the empirical autocorrelations in Figure 3.23. Such data can be described by the methods discussed in Chapter 2. In other words, uncorrelated data are a prerequisite to apply the methods of descriptive statistics discussed in Chapter 2. 

In practice, most of the efficient Monte Carlo techniques generate correlated data. The assumption that a Gaussian approach can stiU be used to calculate the statistical error of correlated data is justified only if it is possible to select a set of uncorrelated data from the original data set. In the rumiing simulation, one literally waits between two measurements imtil the correlation has sufficiently decayed. The effective statistics will be reduced by the waiting time, i.e., the autocorrelation time, which corresponds to the number of sweeps needed to let the correlations decay. If we consider the same total number of sweeps M as in the uncorrelated case, the error will be larger. We expect that it can be conveniently rewritten as... 

With the estimated autocorrelation time, the effective number of uncorrelated data, A/ jj = A//At , can be estimated and eventually be used to calculate the error (4.21) of the average O. This method is certainly not very accurate, but absolutely sufficient to obtain reliable error bars for O and linear functions of it ( the error of the error is typically of no particular interest). 

It can be shown [4] that the innovations of a correct filter model applied on data with Gaussian noise follows a Gaussian distribution with a mean value equal to zero and a standard deviation equal to the experimental error. A model error means that the design vector h in the measurement equation is not adequate. If, for instance, in the calibration example the model was quadratic, should be [1 c(j) c(j) ] instead of [1 c(j)]. In the MCA example h (/) is wrong if the absorptivities of some absorbing species are not included. Any error in the design vector appears by a non-zero mean for the innovation [4]. One also expects the sequence of the innovation to be random and uncorrelated. This can be checked by an investigation of the autocorrelation function (see Section 20.3) of the innovation. 

Fig. 24.2. Single-molecule recording of T4 lysozyme conformational motions and enzymatic reaction turnovers of hydrolysis of an E. coli B cell wall in real time, (a) This panel shows a pair of trajectories from a fluorescence donor tetramethyl-rhodamine blue) and acceptor Texas Red (red) pair in a single-T4 lysozyme in the presence of E. coli cells of 2.5mg/mL at pH 7.2 buffer. Anticorrelated fluctuation features are evident. (b) The correlation functions (C (t)) of donor ( A/a (0) Aid (f)), blue), acceptor ((A/a (0) A/a (t)), red), and donor-acceptor cross-correlation function ((A/d (0) A/d (t)), black), deduced from the single-molecule trajectories in (a). They are fitted with the same decay rate constant of 180 40s. A long decay component of 10 2s is also evident in each autocorrelation function. The first data point (not shown) of each correlation function contains the contribution from the measurement noise and fluctuations faster than the time resolution. The correlation functions are normalized, and the (A/a (0) A/a (t)) is presented with a shift on the y axis to enhance the view, (c) A pair of fluorescence trajectories from a donor (blue) and acceptor (red) pair in a T4 lysozyme protein without substrates present. The acceptor was photo-bleached at about 8.5 s. (d) The correlation functions (C(t)) of donor ((A/d (0) A/d (t)), blue), acceptor ((A/a (0) A/a (t)), red) derived from the trajectories in (c). The autocorrelation function only shows a spike at t = 0 and drops to zero at t > 0, which indicates that only uncorrelated measurement noise and fluctuation faster than the time resolution recorded (Adapted with permission from [12]. Copyright 2003 American Chemical Society)...

If the Monte Carlo updates in each sample are performed completely randomly without memory, i.e., a new conformation is created independently of the one in the step before (which is a possible but typically very inefficient strategy), two measured values Om and 0 are uncorrelated, if m n. Then, the autocorrelation function simplifies to Amn = mn-Thus, the variances of the individual data and of the mean are related with each other by... 

---