Multivariate autocorrelation

A modified Langevin equation can be derived for any property 0t. In addition the memory function will be related to the autocorrelation function of the random force in this equation. These results can be extended to multivariate processes. 

Multivariate autocorrelation of x(t) contains all relationships between the various variables withinx(t) (e.g. x1 x2,. .., x ) and also the relationships ofx(t) which are dependent upon lag. 

The multivariate autocorrelation matrix Rxx %) (for centered values) is defined as ... 

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 ... 

In the multivariate case, the significant cross-correlation or autocorrelation coefficients for each variable add up to the significant multivariate correlation value. 

Because of the highly scattered temporal distributions of the individual loadings of the elements, the multivariate autocorrelation function was computed as described in Section 6.6.3. The results are demonstrated in Fig. 7-1. 

Fig. 7-1. Multivariate autocorrelation function of the sampling interval for suspended dust...

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. 

Fig. 7-18. Multivariate autocorrelation function for the first fraction (dmedium = 8.5 (tm)...

Fig. 7-20. Duration of multivariate autocorrelation as a function of particle size...

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 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... 

The multivariate autocorrelation function of the measured values compared with the highest randomly possible correlation value shows significant correlation up to Lag 7. So, the range of multivariate correlation is more extended than that of univariate correlation (see Section 9.1.3.3.1). This fact must be understood because the computation of the MACF includes the whole data matrix with all interactions between the measured parameters. For characterization of the multivariate heavy metal load of the test area only 14 samples in the screen are necessary. 

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. 

Some statistics concepts such as mean, range, and variance, test of hypothesis, and Type I and Type II errors are introduced in Section 2.1. Various univariate SPM techniques are presented in Section 2.2. The critical assumptions in these techniques include independence and identical distribution [iid) of data. The independence assumption is violated if data are autocorrelated. Section 2.3 illustrates the pitfalls of using such SPM techniques with strongly autocorrelated data and outlines SPM techniques for autocorrelated data. Section 2.4 presents the shortcomings of using univariate SPM techniques for multivariate data. 

L Xie, U Kruger, D Lieftucht, T Littler, Q Chen, and S-Q Wang. Statistical monitoring of dynamic multivariate processes - Part 1. Modeling autocorrelation and cross-correlation. Ind. Engg. Chem. Research, 45 1659—1676, 2006. 

Another limitation of existing SPC methods is that they require the measurements to be uncorrelated, or white, whereas, in practice, autocorrelated measurements are extremely common. A common approach for decorrelat-ing autocorrelated measurements is to approximate the measurements by a time series model, and monitor the residual error. Unfortunately, this approach is not practical, particularly for multivariate processes with hundreds... 

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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