Auto-correlation test

So, as a result, if there is any chaos in ip(t), the auto-correlation test does not show it. We would like to point out, however, that in the semi-classical hmit, i.e. the limit where Planck s constant h is small compared... 

The residuals e, = (y - yt) should also fit a normal distribution, i.e. they should not correlate with xt. An easy way to check this is to plot et versus x . If the residuals do not scatter randomly around zero, the linear model may not be adequate for the data. An indication of a wrongly specified model may be the occurrence of auto-correlated residuals, which can be checked by the DURBIN-WATSON test [MAGER, 1982],... 

Due to the use of a confocal volume, FCS is particularly suited for miniaturization in HTS and relatively insensitive to auto fluorescent test compounds. Moreover, in compound testing, the small path length of the confocal volume greatly limits any filter effects on fluorescence intensity. As in FP, the requirement for large differences in mass in the assay design is a limitation to the applicability of FCS. However, it can be overcome by methods like Fluorescence Intensity Distribution Analysis (FIDA) or two-colour cross correlation derived from the original FCS concept. 

A popular test for the presence of deterministic chaos in ip t) is to examine auto-correlation functions of ip t), for instance. 

Another test for the presence of quantum chaos is to investigate the power spectrum of the auto-correlation function 9 t). It is immediately obvious from (1.4.8) that the power spectrum of a bounded system is a countable sum of S functions. Since no chaos is present in (1.4.8), one may conjecture that for chaos to be present it is necessary that the power spectrum of 9 t) contains a continuous component. Other tests for quantum chaos are to compute the power spectrum of expectation values of dynamical variables, for instance the position x(t) = ip t) x ip t)). Again, a purely discrete power spectrum indicates regular time evolution of 0) whereas a continuous component indicates the presence of chaos. Obviously, a continuous component in the power spectrum is only a necessary condition for chaos. It is not sufficient since unbounded systems, for instance scattering systems, may show a continuous component without any sign of chaos in their dynamics. This state of affairs was known more than 20 years ago (Lebowitz and Penrose (1973), Hogg and Huberman (1982), Wunner (1989)). 

Unfortunately, neither PACF nor ACF lead to directly interpretable results for ARMA processes. The extended ACF tries to overcome this drawback by jointly providing information about the order of both components. For each AR order tested, the EACF first determines estimates of the AR coefficients by a sequence of regression models. Afterwards, the residuals ACF is calculated. The results are presented in a table indicating significant or non-significant auto-correlations (typically denoted by an x and o, respectively). In such a table, the rows represent the AR order p whereas columns represent... 

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... 

Fw sii e exponentkl decay functions with reasonaUy long dec times (> 2 ns) deconvolution poses few difficulties. Double exponential decays, for which C(t) contains 4 variable parameters, are much more difficult to aiuilyse correctly and a number of tests have been proposed to ensure that the fitting procedure does not mask distortions in the data or the presence of a further component. The most common of these tests is a plot of the weighted residuals, which ould be rand( nly distributed about zero. Also of use is a plot of the auto-correlation function of the resid-uals ... 

The auto- and cross-correlation plots are shown in Fig. 6.12. A comparison between the predicted and actual levels is shown in Fig. 6.13. Both figures use the validation data set for testing the model. From Fig. 6.12, it is clear that the residuals are not uncorrelated with each other or the inputs. Therefore, the initial model needs to be improved. Since there is a suggestion that the process model is incorrectly specified, it will first be changed. The best approach is to increase the order of the numerator and denominator (of the B- and F-polynomials) until either the cross-correlation plot shows the desired behaviour or the confidence intervals for the parameters cover zero. If the second case is reached, then this could be a suggestion that a linear model is insufficient/inappropriate for the given data set. Furthermore, the fit between the predicted and measured levels is not great (55.4%). 

To get a high measurability Td, Ta and Tg should be small compared to Tx, and Sa small with respect to Sx. Usually the effect of Tg can be neglected, whereas the value of Ta, which can be simply selected by the operator, is taken equal to Td which is determined by the total analytical system. This means that the next test sample is offered to the analytical system as soon as the analytical result of the previous test sample is available. As can be seen Tx is the key factor with regard to time. It is defined as the time span (DT) over which a reasonable correlation exist between two successive measurements in a time-series (fig.l A). Tx can be evaluated from the auto-covariance function (G(DT) of this time-series Fig.lB)[2,3]. 

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