Correlation time-series

Tab. 6-2. Scheme of cross-correlated time series of metal concentrations in a river...

The value estimated by this way for the overall freshwater DOC concentration is 380 pmol/l C and DOC concentration is reduced by mixing the freshwater inflows with the brackish water of the Baltic Sea by approximately 7.8 pmol/l C, if the salinity increases by 1 psu. It should be kept in mind that this estimate of the freshwater DOC concentration does not take into account any variability of riverine runoff, seasonal variability of DOC concentrations in the inflowing freshwater or atmospheric precipitation, which is thought to be essentially free of natural organic compounds. Using this correlation, time series data for DOC can be corrected for the background concentration as well as for the variability caused by salinity changes. For example, a corrected time series of DOC in the surface layer of station 271 is shown in Fig. 12.17. 

In Aviv (2002a), we extend the model of Aviv (2001) by specifically considering questions related to VMI and CPFR, and by using an auto-correlated time series framework for the underlying demand process (we shall provide... 

The inventory management literature (Clark and Scarf 1960) show that an echelon-based policy is optimal for a centrally-managed supply chain. However, no optimality results where demonstrated for a general linear state space demand model, in which different echelons observe different information about the demand, and the demand itself follows a correlated time-series pattern. Nevertheless, there are some interesting and relevant works related to supply... 

When experimental data are collected over time or distance there is always a chance of having autocorrelated residuals. Box et al. (1994) provide an extensive treatment of correlated disturbances in discrete time models. The structure of the disturbance term is often moving average or autoregressive models. Detection of autocorrelation in the residuals can be established either from a time series plot of the residuals versus time (or experiment number) or from a lag plot. If we can see a pattern in the residuals over time, it probably means that there is correlation between the disturbances. 

The correlation of values within a time series plays an important role in sampling. It can be characterized by means of the autocorrelation function (Doerffel and Wundrack [1986], Chatfield [1989], Hartung et al. [1991])... 

The key to calculating the Durbin-Watson statistic is that prior to performing the calculation, the data must be put into a suitable order. The Durbin-Watson statistic is then sensitive to serial correlations of the ordered data. While the serial correlation is often thought of in connection with time series, that is only one of its applications. 

The correlation coefficients between a 10 year monthly mean time series of volatilisation rates and SST, 1 Om wind speed and pollutant concentration are used to elucidate which of the parameters drives the volatilisation rate changes and causes the deviations from the long term mean. All of the parameters do not vary independently. Since both SST and wind speed influence the volatilisation rate in a nonlinear manner, it is not intuitive whether an increase in wind speed leads to an increase in volatilisation rate. A raise in wind speed that coincides with a decrease of the sea surface temperature can lead to a negative linear correlation coefficient between volatilisation rate and wind speed. For that reason the partial correlation coefficient is calculated in addition to the simple linear correlation coefficients. It explains the relation between a dependent and one or more independent parameters with reduced danger of spurious correlations due to the elimination of the influence of a third or fourth parameter, by holding it fixed. One important feature of the partial correlation coefficient is, that it is equal to the linear correlation coefficient if both variables... 

Autocorrelations were calculated on the time-series of tongue-flick duration scores during earthworm extract trailing. This was done for a series of tongue-flick-number shifts. So, for example, for a shift of 1 the autocorrelation paired each tongue-flick duration with the subsequent tongue flick duration. For a shift of two, the correlation paired each tongue flick duration with the second to follow. 

The mobility of tyrosine in Leu3 enkephalin was examined by Lakowicz and Maliwal/17 ) who used oxygen quenching to measure lifetime-resolved steady-state anisotropies of a series of tyrosine-containing peptides. They measured a phase lifetime of 1.4 ns (30-MHz modulation frequency) without quenching, and they obtained apparent rotational correlation times of 0.18 ns and 0.33 ns, for Tyr1 and the peptide. Their data analysis assumed a simple model in which the decays of the anisotropy due to the overall motion of the peptide and the independent motion of the aromatic residue are single exponentials and these motions are independent of each other. 

The time-series analysis results of Merz et were expressed in first-order empirical formulas for the most part. Forecasting expressions were developed for total oxidant, carbon monoxide, nitric oxide, and hydrocarbon. Fitting correlation coefficients varied from 0.547 to 0.659. As might be expected, the best results were obtained for the primary pollutants carbon monoxide and nitric oxide, and the lowest correlation was for oxidant. This model relates one pollutant to another, but does not relate emission to air quality. For primary pollutants, the model expresses the concentrations as a function of time. 

Structural information on aromatic donor molecule binding was obtained initially by using H NMR relaxation measurements to give distances from the heme iron atom to protons of the bound molecule. For example, indole-3-propionic acid, a structural homologue of the plant hormone indole-3-acetic acid, was found to bind approximately 9-10 A from the heme iron atom and at a particular angle to the heme plane (234). The disadvantage of this method is that the orientation with respect to the polypeptide chain cannot be defined. Other donor molecules examined include 4-methylphenol (p-cresol) (235), 3-hydroxyphenol (resorcinol), 2-methoxy-4-methylphenol and benzhydroxamic acid (236), methyl 2-pyridyl sulfide and methylp-tolyl sulfide (237), and L-tyrosine and D-tyrosine (238). Distance constraints of between 8.4 and 12.0 A have been reported (235-238). Aromatic donor proton to heme iron distances of 6 A reported earlier for aminotriazole and 3-hydroxyphenol (resorcinol) are too short because of an inappropriate estimate of the molecular correlation time (239), a parameter required for the calculations. Distance information for a series of aromatic phenols and amines bound to Mn(III)-substituted HRP C has been published (240). 

However, in other types of molecular simulation, any sampled configuration may be correlated with configurations not sequential in the ultimate "trajectory7 produced. That is, the final result of some simulation algorithms is really a list of configurations, with unknown correlations, and not a true trajectory in the sense of a time series. 

Methods based on the MVN distribution have been used particularly for autocorrelated data, for example, in time series analysis and geostatistics. Autocorrelation occurs when the same variable is measured on different occasions or locations. It often happens that measurements taken close together are more highly correlated than measurements taken less close together. Environmental data often have some type of autocorrelation. 

To get better insight into the importance of temperature selection, we have recorded a series of NOESY spectra of cyclo(Pro-Gly) at different temperatures. The dashed lines in fig. 6 show theoretical dependence of the laboratory frame cross-relaxation rate on the correlation time (according to... 

Continued) Distances were obtained from cross-relaxation rates from eq. (33a) with Tc = 4.0 ns and ujo/l-n = 500 MHz correlation time was determined from eq. (la) using GlyH /H spin pair a = —7.22 s (from build up rate analysis of a series of spectra, table 4) r = 1.77 A (from model, table 1). Interproton error limits were obtained from the errors of respective cross-relaxation rates Ar/r = Ao /cr /6. 

In an attempt to address these questions, a modern method of statistical physics was recently applied by Varotsos et al. (2007) to C02 observations made at Mauna Loa, Hawaii. The necessity to employ a modern method of C02 data analysis stems from the fact that most atmospheric quantities obey non-linear laws, which usually generate non-stationarities. These non-stationarities often conceal existing correlations between the examined time series and therefore, instead of applying the conventional Fourier spectral analysis to atmospheric time series, new analytical techniques capable of eliminating non-stationarities in the data should be utilized (Hu et al., 2001 Chen et al., 2002 Grytsai et al., 2005). 

According to Varotsos et al. (2007) we begin the analysis of the time series (shown in Figure 3.13) by investigating whether the C02 concentration at different times is actually correlated. The motivation for this investigation stems from the observation that many environmental quantities have values which remain residually correlated with one another even after many years (long-range dependence). 

The aim of correlation analysis is to compare one or more functions and to calculate their relationship with respect to a change of t (lag) in time or distance. In this way, memory effects within the time curve or between two curves can be revealed. Model building for the time series is easy if information concerning autocorrelation is available. 

The autocorrelation function of autocorrelated time series (first or higher order) has an exponentially decreasing shape. An autocorrelation of the order unity means only a correlation between x(t) and x(t— 1). Because of the same correlation between x(t - 1) to x(t - 2) it seems that a correlation between x(t) and x(t - 2) is also detectable. For this reason, it is very difficult to find the correct order of the autocorrelation process. A useful tool in this case is the partial correlation coefficient. 

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 ... 

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