Autocorrelation time lags

Since the time series of only one variable is involved and the time lag / between the two time series is the parameter that changes, the autocorrelation coefficient is represented by r . The upper limit of the summation in the denominator varies with 1. In order to have an equal number of data in both series, n — I values are used in the summation. 

Figures 9.24-9.26 show, respectively, the cross-correlations, autocorrelations, and power spectrum of the detector signals for the 500-pm glass particles at = 2. In Fig. 9.24, the curves for the opposite detector pairs, 9/11 and 10/12, are of particular interest. They exhibit nearly zero values of correlation at zero time lag, indicating significant antisymmetric sloshing motion. The autocorrelations, shown in Fig. 9.25, reveal the existence of both near-periodic and random fluctuations. The power spectrum of the detector signals in Fig. 9.26 shows the dominant fluctuations of the solids motion in the bed. The dominant frequencies for the 500- and 705-mm glass particles at u,/umf = 1.5, 2, 3, and 4 are listed in Table 9.2. The...

Figure 7.11(a) shows the results from a scheme in which all three groups of rate constants are chosen to be slow (no rate-determining step). When all the rate coefficients in a set of consecutive reactions are comparable, the stationary state hypothesis is often employed to solve analytically the kinetic equations. The approximate autocorrelation time for the time series, defined as the time lag at which the correlation of a series with itself decays to zero, is about 2.5 s (50 lag times, i.e., 50 observations). The system is not very close to steady state in this case, but a reasonably linear MDS diagram is produced nonetheless. There are three points to be made ... 

As a measure of the degree of correlation, the empirical autocorrelation is applied (cf. correlation coefficient according to Eq. (5.12)). For autocorrelation of a function of n data points, the empirical autocorrelation, r(r), for time lag t is defined by... 

In order to evaluate all possible autocorrelations, r(r) is plotted against the time lag t in a correlogram, also called an autocorrelation function. The autocorrelation function for the time series of sulfur concentrations in snow is given in Figure 3.22. The time lags T correspond here to monthly distances. 

The measurement at time t plus the time lag r is predictable on the basis of the autocorrelation coefficient, r(r), and the y value at time t. Here, e represents the random error. Note that this is a very simple model. Good predictions can only be expected for time series that really obey this simple autoregressive model. 

This is a time (lag) domain function that expresses the similarity of a signal to lagged versions of itself. It can also be viewed as the convolution of a signal with the time-reversed version of itself. Pure periodic signals exhibit periodic peaks in the autocorrelation fimction. Autocorrelations of white noise sequences exhibit only one clear peak at the zero lag position, and are small (zero for infinite length sequences) for all other lags. 

Remember, we used autocorrelation as a means to do pitch estimation in Chapter 5. Autocorrelation computes a time (lag) domain function that... 

The selection of the time delay t is done in such a way that it makes every component of phase space uncorrelated. Therefore, r is determined from estimate of the autocorrelation function of the time series. The time lag that corresponds to the first zero in the autocorrelation is often chosen as a good approximation for t. - The determination of D2 in practice can be done using the Grass-berger-Procaccia algorithm outlined below. Consider a pair of points in space with m dimensions (m < k) at time instants i and j ... 

Time lag analysis of restoration processes (autocorrelations and crosscorrelations of restoration curves)... 

Fig. 7 PC-AR(6) model validation results, (a) The autocorrelation function for the prediction error sequence of the validation set (simulation experiment 2 time lags 100 95 % confidence limits) and (b) the covariance matrix of the prediction error matrix E...

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. 

The aim of correlation analysis is to compare one or more functions and to calculate their relationship with respect to a change of t (lag) in time or distance. In this way, memory effects within the time curve or between two curves can be revealed. Model building for the time series is easy if information concerning autocorrelation is available. 

A random process is weakly stationary if its mean value and autocorrelation function are independent of r. Thus, for a weakly stationary random process, the mean value is a constant [fJiy r) = fiy] and the autocorrelation function depends only on the spatial lag 6 [e.g., Ryir, r + 6) = Ry d)]. A random process is strongly stationary if the infinite collection of higher order statistical moments and joint moments are space invariant. Most geophysical phenomena are not strongly stationary. However, the random process under study must be at least weakly stationary, otherwise the results of the space- or time-series analysis can be suspect. An extensive treatment of these statistical concepts is available 45, 46). A detailed re-... 

To decide on the order of time series models as well as to check residuals for the white noise assumptions, auto-correlations of time series and residuals need to be calculated and analysed. Standard metrics to analyse for time-dependent correlation structures are the autocorrelation function (ACF), partial ACF (PACF) and extended ACF (EACF). The ACF estimates the empirical auto-correlations between lagged observations... 

While the estimates of the autocorrelation coefficients for the Cg time series (lower rows in 1 to ordy change slightly, the estimates the autocorrelation coefficients for the Benzene time series (upper rows in to 3) are clearly affected since three parameters are dropped from the model. The remaining coefficients are affected, too. In particular, the lagged cross-correlations to the Cg time series change from 1.67 to 2.51 and from -2.91 to -2.67 (right upper entries in 1 and This confirms the serious effect of even unobtrusive outliers in multivariate times series analysis. By incorporating the outliers effects, the model s AIC decreases from -4.22 to -4.72. Similarly, SIC decreases from -4.05 to -4.17. The analyses of residuals. show a similar pattern as for the initial model and reveal no serious hints for cross- or auto-correlation. i Now, the multivariate Jarque-Bera test does not reject the hypothesis of multivariate normally distributed variables (at a 5% level). The residuals empirical covariance matrix is finally estimated as... 

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