Time-lagged correlation function

From the measured (in this case calculated) time series of the concentrations in fig. 7.2 we form correlation functions. Denote by x,- (f) the concentration of species I, at time t, and by (x,) the average of that concentration over the entire time of the measurements then the time-lagged correlation function of two species i and j, Sij (t), is defined by... 

Fig. 8.3 (A) Plot of the time-lagged correlation function R of G6P with all other species. Each lag represents a time of 13 min.

Panasenkov (18C) has studied the influence of turbulence of the liquid jet on its atomization. With the onset of turbulence the jet becomes shorter and begins to break up. He correlates the drop diameter as a function of the orifice diameter and the Reynolds number. Miesse (12C, 13C) has made a careful investigation of the effect of time lag of jet breakup on combustion stability and has correlated the influence of ambient pressure oscillations on the liquid jet. 

The use of the cross-correlation function enables not only the determination of the relationship between two different time series, x(t) and y(t), as the ordinary correlation coefficient does, but also in relation to a time lag ... 

The time series example used in this section requires one to determine the time lag between the storage reservoir feeder stream and the drinking water storage reservoir. The cross-correlation function was then calculated between the nitrate concentrations in the... 

The storage reservoir and the feeder stream both show the order one for autoregression, but a time lag of two months between the two series is detected by the cross-correlation function (Fig. 6-15). For this reason, it is necessary to modify the general model to ... 

Another method of particle velocity measurement is the double tip optical fiber, which consists of two groups of optical fibers with a certain distance between each other, as shown in Fig. 6. When a solid particle passes the probe tip, two signals with a certain time interval in between would be detected. This time lag can be computed by means of the cross-correlation function (Qin and Liu, 1982) ... 

In order to obtain the time lag between two sets of reflected light signals of known distance apart, points a and b, for particles in random movement, the cross-correlation function method is generally used in treating the signals. The cross-correlation function of two sets of random signals a t) and b(t) express the independence of the two sets of sampled data, i.e.,... 

Under the same reaction conditions, the equilibrium method can determine Ame for ethanol below 0.20 mmol/L after reaction for 50 min. For kinetic analyses of such reaction curves, the lag time for steady-state reaction is estimated to be over 40 s and is used to select data of steady-state reaction for analysis. Using the equilibrium method as the reference method, the best steady-state Caid for data of 6.0-min reaction is obtained for consistency of A k with Ame at each tested ethanol level from 10 tmol/L to 0.17 mmol/L. After dilution and determination by the equilibrium method, Ame for each tested ethanol level from 0.17 mmol/L to 0.30 mmol/L is also available. Consequently, an exponential additive function is obtained to apvproximate the correlation of the best Caid for predicting Amk consistent with Ame (Fig. 9). This special correlation function for Caid and Amk is used as a restriction function to iteratively adjust Caid for predicting Amk namely, iterative kinetic analysis of reaction curve with Caid predicted from the restriction function using previous Amk finally gives the desired Amk- Such an artificial intelligence approach to the steady-state Caid for kinetic analysis of reaction curve can hardly be found in publications. 

As a measure of the degree of correlation, the empirical autocorrelation is applied (cf. correlation coefficient according to Eq. (5.12)). For autocorrelation of a function of n data points, the empirical autocorrelation, r(r), for time lag t is defined by... 

To decide on the order of time series models as well as to check residuals for the white noise assumptions, auto-correlations of time series and residuals need to be calculated and analysed. Standard metrics to analyse for time-dependent correlation structures are the autocorrelation function (ACF), partial ACF (PACF) and extended ACF (EACF). The ACF estimates the empirical auto-correlations between lagged observations... 

Rq is a 3-array of the correlation function Rg where the first two indices % denote the modes and the last denotes the time lag... 

Fig. 5 Van Hove correlation functions for the ensemble-average particles, at time lag 1 s, moving in glycerol 85% (A), Sterocoll FD 1% (B), Sterocoll D 1% (C). The ensemble-average data is fit by a Gaussian distribution, a value is a measure of the fit of the curve to the data, with 0 being a perfect fit...

Now let s see how (u(O)u(t)), a property of the fluctuations in equilibrium, is related to a kinetic property, in this case the diffusion coefficient D. Integrate the time correlation function Equation (18.68) over all the possible time lags t. 

The time shift r is referred to as the correlation time. The cross correlation function is a measure of the close matching or similarity of functional forms or patterns between the two signals. As the time axis of the signal measured at V2 is shifted forward by t, the development of a significant value for C(t) indicates that the matched portion of the signal measured at V2 is lagging behind that measured at Vj by t. 

First, a distinction can be made between non-parametric and parametric identification. Non-parametric system identification involves the estimation of an impulse response function, frequency response function (FRF), correlation function, or power spectral density (PSD), not as a mathematical function depending on a few parameters, but as a set of tabulated values for each considered time lag or frequency. Although nonparametric models are sometimes directly used for modal analysis, they are most often used as preprocessed data for parametric identification since the estimation accuracy of parametric approaches is much higher than that of nonparametric approaches (Peeters and De Roeck 2001 Reynders 2012). 

---