Univariate Autocorrelation Analysis

Mathematical methods for calculating correlation are applied to describe the degree of relationship between one or more measuring rows (for mathematical fundamentals see Section 6.6). The theoretical fundamentals of univariate auto- and cross-correlation ana- 

By analogy with the procedure by DOERFFEL et al. [1988] for testing homogeneity in analytical investigations of solids, the regular raster screen of the investigated area is divided into a meandering shape, i.e. the two-dimensional local series is transformed into a one-dimensional local series. 

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

The autocorrelation functions of the investigated features calculated according to Eq. 6-22 and smoothed according to Eq. 9-4 are represented in Fig. 9-5. The increasing distance l of the measuring points results in steadily decaying autocorrelation functions for all features, with that for lead decaying most conspicuously. 

The points of intersection between the highest value of a random correlation and the lower limit of the confidence interval of the corresponding empirical model function according to Eq. 9-4 correspond to the lower limits of the confidence range of the critical distances between the sampling points. These values are represented in Tab. 9-2. 

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The multivariate autocorrelation function should contain the total variance of these autocorrelation matrices in dependence on the lag x. Principal components analysis (see Section 5.4) is one possibility of extracting the total variance from a correlation matrix. The total variance is equal to the sum of positive eigenvalues of the correlation matrices. This function of matrices is, therefore, reduced into a univariate function of multivariate relationships by the following instruction ... 

It is assumed that the residuals are independent, normally distributed, with mean zero and constant variance. These are standard assumptions for maximum likelihood estimation and can be tested using standard methods examination of histograms, autocorrelation plots (ith residual versus lag-1 residual), univariate analysis with a test for normality, etc. 

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