Multivariate Autocorrelation Analysis

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

The multivariate autocorrelation function (MACF) was computed for each particle-size fraction of the data set, consisting in the concentrations of 23 elements in aerosol samples taken weekly over a period of 60 weeks. 

In the first fraction (dmedium = 8.5 pm) significant autocorrelation can be found after a lag of one week only (Fig. 7-18). Because of the rapid deposition of these large particles there is no longer-term correlation with impact of the emissions. 

In comparison with Fig. 7-18 the MACF for the fifth fraction (dmedium = 0.45 pm) (Fig. 7-19) is less scattered and the duration of correlation is definitely longer - significant multivariate correlation with the impact of the emissions can be found, for up to 12 weeks. 

This tendency of increasing duration of correlation with decreasing particle size is illustrated in Fig. 7-20. This means, that the impact of smaller particles is more uniform than that of larger particles, which are deposited relatively quickly. Thus it is possible to characterize the long-term impact level of smaller airborne dust particulates. 

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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 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. 

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 ... 

The computation of the multivariate autocorrelation function (MACF) is useful if the simultaneous consideration of all measured variables and their interactions is of interest. The mathematical fundamentals of multivariate correlation analysis are described in detail in Section 6.6.3. The computed multivariate autocorrelation function Rxx according to Eqs. 6-30-6-37 is demonstrated in Fig. 9-6. The periodically encountered... 

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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