Autocorrelation correlated data

Figure 6. Autocorrelation function data at 6 = 24° (+) for LLS sample 3. Generated time correlation function (continuous curve) from the size distribution curve of TEB as shown in figure 4 with relative deviation between the two curves. Deviation = 1.0 - l Ogenerated Si measured ...

The strength of correlation dies out as the number of sampling intervals between observations increases. In other words, as the sampling interval increases, the correlation between successive samples decreases. In some industrial monitoring systems, a large sampling interval is selected in order to reduce correlation. The penalty for this mode of operation is loss of information about the dynamic behavior of the process. Such policies for circumventing the effects of autocorrelation in data should be avoided. 

To characterize correlated data by autocorrelation and cross-correlation functions. 

Correlated Data A random time series with low correlations between observations provides an autocorrelation function as shown in Figure 3.24. In the case of a stationary process of the first order, the function can be described by the following exponential model ... 

Figure 3.24 Autocorrelation function for weakly correlated data according to a process of first order. The time series is based on glucose determinations in urine over time.

Two approaches are typically followed in this case. The first approach involves the analysis of the underlying statistical model of the autocorrelation (e.g., ARIMA model) and the monitoring of the residuals with traditional control charts for independent observations. The second approach develops monitoring schemes directly on the correlated data (e.g., EWMA control charts). 

Schatzel, K., Noise on Photon Correlation Data I. Autocorrelation Functions, Quantum Opt, 1990, 2, 287-306. 

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... 

When experimental data are collected over time or distance there is always a chance of having autocorrelated residuals. Box et al. (1994) provide an extensive treatment of correlated disturbances in discrete time models. The structure of the disturbance term is often moving average or autoregressive models. Detection of autocorrelation in the residuals can be established either from a time series plot of the residuals versus time (or experiment number) or from a lag plot. If we can see a pattern in the residuals over time, it probably means that there is correlation between the disturbances. 

Here t is a correlation length, which grows as correlation in the data grows. The net effect of t is to reduce the effective number of independent data points, t is calculated from the autocorrelation coefficients for the series of data ... 

Methods based on the MVN distribution have been used particularly for autocorrelated data, for example, in time series analysis and geostatistics. Autocorrelation occurs when the same variable is measured on different occasions or locations. It often happens that measurements taken close together are more highly correlated than measurements taken less close together. Environmental data often have some type of autocorrelation. 

The method most commonly used is an autocorrelation analysis.46 A set of data collected are divided into n subsets of the same size. The correlation coefficient R(k) is calculated according to... 

In our approach [1, 2] termed the dynamic method the complex susceptibility x = x — ix" is determined by a law of undamped motion of a dipole in a given potential well and by dissipation mechanism often described as stosszahlansatz in the underlying kinetic or Boltzmann equation. In this review we shall refer to this (dynamic) method as the ACF method, since it is actually based on calculation of the spectrum of the dipolar autocorrelation function (ACF). Actually we use a one-particle approximation, in which the form of an employed potential well (being in many cases rectangular or close to it) is taken a priori. Correlation of the particles coordinates is characterized implicitly by the Kirkwood correlation factor g, its value being taken from the experimental data. The ACF method is simple and effective, because we do not employ the stochastic equations of motions. This feature distinguishes our method from other well-known approaches—for example, from those described in books [13, 14]. 

The multivariate autocorrelation function of the measured values compared with the highest randomly possible correlation value shows significant correlation up to Lag 7. So, the range of multivariate correlation is more extended than that of univariate correlation (see Section 9.1.3.3.1). This fact must be understood because the computation of the MACF includes the whole data matrix with all interactions between the measured parameters. For characterization of the multivariate heavy metal load of the test area only 14 samples in the screen are necessary. 

The autocorrelation (or correlation) function is obtained by multiplying each y (f) by y (t — t°), where t° is a time delay, and summing the products over all points [43]. Examination of the sum plotted as a function of t° reveals the level of dependency of data points on their neighbors. The correlation time is the value of t° for which the value of the correlation function falls to exp (—1). When the correlation function falls abruptly to zero, that indicates that the data are without a deterministic component a slow fall to zero is a sign of stochastic or deterministic behavior when the data slowly drop to zero and show periodic behavior, then the data are highly correlated and are either periodic or chaotic in nature [37,43]. 

Finally, we comment on the difference between the self part and the full density autocorrelation function. The full density autocorreration function and the dynamical structure factor ire experimentally measured, while in the present MD simulation only the self pairt was studied. However, the difference between both correlation functions (dynamical structure factors) is considered to be rather small except that additional modes associated with sound modes appear in the full density autocorrelation. We have previously computed the full density autocorrelation via MD simulations for the same model as the present one, and found that the general behavior of the a relaxation was little changed. General trends of the relaxation are nearly the same for both full correlation and self part. In addition, from a point of numerical calculations, the self pMt is more easily obtained than the full autocorrelation the statistics of the data obtained from MD simulatons is much higher for the self part than for the full autocorrelation. 

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)...

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