Autocorrelation analysis

Similar models have been applied in geological exploration and environmental studies. There semivariogram analysis (Akin and Siemens [1988], Einax et al. [1997]) plays a comparable role than autocorrelation analysis for the characterization of stochastic processes. 

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... 

BROWNIAN MOTION AND AUTOCORRELATION ANALYSIS OF SCATTERED LIGHT INTENSITY... 

Objects which are internally correlated for example volumes sampled from rivers, soils, or ambient air, can be treated by autocorrelation analysis or semivariogram analysis. The range up to a critical level of error probability is an expression of the critical spatial or temporal distance between sampling points. 

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],... 

When applying multivariate autocorrelation analysis to this multivariate problem (for mathematical fundamentals see Section 6.6.3) two questions should be answered ... 

To investigate the influence of wind direction, the factor scores for each fraction were averaged within a sector of 30°. The graphical representation of the scores of both factors (computed from the data set of the first fraction) versus the angle of wind direction is very noisy (Fig. 7-21) and does not enable any conclusions to be drawn on the location of these emission sources. This result is in good agreement with the result from multivariate autocorrelation analysis of the first fraction. 

In accordance with this fact and also with the result from multivariate autocorrelation analysis, the factor scores for smaller particles depend on wind direction. This dependence is illustrated by the example of the fifth fraction of particles in Fig. 7-23. The factor scores of the first, anthropogenic, factor have a broad maximum in the range of 130-180°. Comparison with the frequency distribution of wind direction in the time interval under investigation (Fig. 7-24) shows that the direction in which the scores of this anthropogenic factor have a maximum (Fig. 7-23) does not correspond with the most frequent wind direction (240-330°). This maximum of factor scores in the range of 130-180° indicates the influence of industrial and communal emissions in the conurbations of Bremen and Hamburg. 

The determination of the critical distance between the sampling points can then be performed by autocorrelation analysis on the analytically determined concentrations. 

The application of multivariate autocorrelation analysis is useful for determination of the distance between samples for representative sampling for characterization of multivariate loading by heavy metals. 

Spatial autocorrelation analysis was used to quantify patterns in two-dimensional microscale distribution of chlorophyll a concentrations (pgChl-a l-1) and relative seawater excess viscosity rj (%). The objective is to link these parameters to a common patch size to quantify their spatial linkage. 

The results of spatial autocorrelation analysis showed that none of the investigated patterns were uniform nor random (Table 4), indicating the existence of structural complexity in 2D microscale patterns of chlorophyll a concentration and seawater viscosity. Except on June 20, consistent spatial patterns were found for chlorophyll and excess... 

The fluorescence fluctuations measured by FCS can be analyzed in several ways. The most common technique, autocorrelation analysis, provides information about characteristic diffusion time of fluorescent molecules through the observation volume. It also reports on the average number of molecules present in the observation volume, and thus the concentration of fluorescent moleculesn (14, 49, 56, 57). Other types of FCS analysis can be used to analyze molecular brightness and the oligomeric state of the fluorescent molecule. Finally, cross-correlation FCS monitors fluctuations jointly from molecules labeled with two or more different fluorophores. This technique provides a powerful approach to assay for intermolecular interactions, because molecules that are bound either directly or indirectly to one another should diffuse as a single unit (8, 59). 

Figures 6 and 7 (see color plate for Fig. 7) show two uses we have made of the autocorrelation function to quantitate features of images. The micrographs in Fig. 6 show grazing optical sections of nuclei from a Drosophila early embryo and a HeLa cell stained with an antinuclear lamin antibody. Using an autocorrelation analysis of images like this, we demonstrated that the mean spacing between adjacent local maxima in the antilamin staining pattern was —0.5 ptm in four...

Spatial autocorrelation analysis of allozyme frequencies led to similar results (Barbujani and Sbordoni, unpublished). With the exception of the previously discussed Pgm locus, a substantial lack of spatial autocorrelation was revealed, suggesting the absence of either constant gene flow, even very low, or selection pressures producing a geographic structure. 

L. J. Bruner and J. E. Hall, Autocorrelation Analysis of Hydrophobic Ion Current Noise in Lipid Bilayer Membranes, Biophys. J. 28, 511-514 (1979). 

Time domain autocorrelation analysis provides a measure of the self-similarity of a time series signal and the decay of the autocorrelation function describes the temporal persistence of information carried by it. The normalized fluorescence correlation function of a fluctuating signal F t) can be written as [29],... 

Higher order autocorrelation analysis is analogous to the analysis of the higher order moments of the photon count distribution. Such a correlation function is given by. 

The normalized data is summarized in Table I. The maxima observed were also confirmed in the autocorrelation analysis which is included in Figure 8. The positions of these maxima were not altered after blocking of the cavity with 1,8-ANS and subsequent removal by flushing with water [26,27]. 

Figure 8. Histogram of individual pull-offforces measured in AFMforce distance curves between a SAM of p-CD 1 and a mixed hydroxyl-ferrocene tip in water (A), after presence of 1,8-ANS and thorough flush with water (B) (arrows indicate force maxima). The autocorrelation analysis of the data obtained in three independent experiments is shown in C [32]. (Reproduced with permission from reference 11. Copyright 2000 American Chemical Society.)...

The original 2D autocorrelation analysis calculates a vector based on the distances between all atoms of a structure and any property of these atoms.205>206 For each pair of atoms, the distance between the atoms (number of bonds between them) and the product of the properties is noted. Each element of the autocorrelation vertor is the sum of these products for one particular distance. A separate autocorrelation vector is calculated for each property of interest—typically, volume, electronegativity, hydrogen bonding character, hydrophobicity. As a final step a principal components analysis reduces the number of variables to consider. 

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