Autocorrelation empirical

The first attempt to account for the structure of the empirically determined velocity autocorrelation function using the memory function... 

The required distance has to be chosen from the empirical semivariogram or (if the first sampling was done equidistantly) by autocorrelation analysis also (see example for soil sampling in Sections 9.1 and 9.4). Clearly, the required distance depends on the relationship between nugget effect and sill. The length determination is described in detail by YFANTIS et al. [1987],... 

The smoothing of the autocorrelation functions is performed by means of regression according to the empirical function ... 

Autocorrelation in data affects the accuracy of the charts developed based on the iid assumption. One way to reduce the impact of autocorrelation is to estimate the value of the observation from a model and compute the error between the measured and estimated values. The errors, also called residuals, are assumed to have a Normal distribution with zero mean. Consequently regular SPM charts such as Shewhart or CUSUM charts could be used on the residuals to monitor process behavior. This method relies on the existence of a process model that can predict the observations at each sampling time. Various techniques for empirical model development are presented in Chapter 4. The most popular modeling technique for SPM has been time series models [1, 202] outlined in Section 4.4, because they have been used extensively in the statistics community, but in reality any dynamic model could be used to estimate the observations. If a good process model is available, the prediction errors (residual) e k) = y k)—y k) can be used to monitor the process status. If the model provides accurate predictions, the residuals have a Normal distribution and are independently distributed with mean zero and constant variance (equal to the prediction error variance). 

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... 

For the time series of sulfur concentrations in snow (Figure 3.20), the empirical autocorrelation is calculated... 

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. 

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... 

Autocorrelation decay functions calculated on the basis of Eq. (44) for different values of the intermoiecular coupling parameter y arc presented in Fig. 4. The ordinate is log[ — In M (x)] and the abscissa is Iog(xX abdi taken as unity. This choice of the coordinates allows direct com rison with the empirical stretched exponential or the Kohlrausch-Williams-Watt (KWW) function [93, 94]... 

One procedure to aid this decision is to plot the logarithm of the different values Sj as a function of 1 (see example later). The plot shows an abrupt change in slope that can be used as a cut-off. Another method is the use of autocorrelation coefficients for the columns of U and V. This coefficient is obtained by summing over the whole column the products of x, and x, +1, where Xj is one of the elements of the column of U (or V), formally c(U) = SUjfUj- +, .j from j = 1 to n - 1. Since the matrices U and V are orthogonal, the values of the elements contained in their columns lie between 1 and -1. According to an empirical rule, columns vrith autocorrelation coefficients with values >0.5 are more likely to contain real signal rather than noise. 

Schriever and co-workers - have published a series of papers on H and D relaxation in poIy(methacrylic acid) solutions. In the first two papers, methylene D Tj, and data and methyl H and carboxylic D Ti and data were investigated as a function of frequency, and the results interpreted empirically in terms of a biexponential autocorrelation function. As a function of d ee of neutralization (x ), the time constants of both exponential terms decrease sharply from a = 0 to 0.5, but thereafter remain constant. Similar conclusions were drawn from methyl linewidths and D solvent relaxation in D2O solutions. ... 

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