Time series modeling model structures

Plot data and autocorrelation functions, postulate structure of time series model,... 

AC Hahn. Forecasting, Structural Time Series Models and the Kalman Filter. Cambridge University Press, New York, NY, 1989. 

To model local chemical production processes at production sites, time series models are used which are able to accurately represent the dynamic, time-dependent structure of chemical production processes. 

The previous section points out which structure of a time series model to choose depending on the general type of the production plant to be modelled. Once a model class is selected, the corresponding model s parameters have to specified based on historical records of the investigated plant. The model fitted to the historical records has then to be checked for adequacy, i.e. the residuals of the model have to be checked for the assumptions inherent to the chosen model (typically white noise). 

The SIC is deduced from Bayesian arguments. It consistently estimates the true order of ARMA(p, q) processes and is probably the most widely used information criterion in univariate time series analysis. The HQIC is the most recent IC and especially designed for multivariate time series models. In practice, multiple ICs are simultaneously calculated which allows the analyst to cross-check the recommendations of the various ICs. Strongly deviating recommendations may indicate an inappropriate model structure. 

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... 

Harvey AC (1991) Forecasting, structural time series models and the Kalman filter. Cambridge University Press, Cambridge... 

In this section, we present an iterative algorithm in the spirit of the generalized least squares approach (Goodwin and Payne, 1977), for simultaneous estimation of an FSF process model and an autoregressive (AR) noise model. The unique features of our algorithm are the application of the PRESS statistic introduced in Chapter 3 for both process and noise model structure selection to ensure whiteness of the residuals, and the use of covariance matrix information to derive statistical confidence bounds for the final process step response estimates. An important assumption in this algorithm is that the noise term k) can be described by an AR time series model given by... 

The results confirm that a structural time series model with explanatory and intervention variables is an appropriate tool for explaining the changes in the monthly number of fatalities in Poland for the period 1998-2012, in relation to economic factors such as the industrial production index and/or the unemployment rate. A prehminary graphical analysis was conducted, which confirmed that the correlation between the munber of fatalities and the industrial production index (and the imemployment rate respectively) was positive (and negative respectively) on average. Log-log and log-lin specifications were then tested for accounting for these correlations, and three models which confirm this average relation were finally retained as statistically satisfactory and interpretable. 

HAR 86] Harvey A.C., Durbin J., The effects of seat belt legislation on British road casualties a case study in structural time series modeling. Journal of the Royal Statistical Society, vol. 149, part 3, pp. 187-227,1986. 

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. 

CO2 production time series, 39 88 equation structure, 39 87-88 Kurtanjek s mechanism, 39 91 oxide models, 39 89-92 subsurface oxygen model, 39 90-91 selective, 30 136-137 small organic molecules, chemical identity of adsorbed intermediates, 38 21 states... 

When a spring and a dash pot are connected in series the resulting structure is the simplest mechanical representation of a viscoelastic fluid or Maxwell fluid, as shown in Fig. 3.10(d). When this fluid is stressed due to a strain rate it will elongate as long as the stress is applied. Combining both the Maxwell fluid and Voigt solid models in series gives a better approximation for a polymeric fluid. This model is often referred to as the four-parameter viscoelastic model and is shown in Fig. 3.10(e). Atypical strain response as a function of time for an applied stress for the four-parameter model is found in Fig. 3.12. 

Using time-resolved crystallographic experiments, molecular structure is eventually linked to kinetics in an elegant fashion. The experiments are of the pump-probe type. Preferentially, the reaction is initiated by an intense laser flash impinging on the crystal and the structure is probed a time delay. At, later by the x-ray pulse. Time-dependent data sets need to be measured at increasing time delays to probe the entire reaction. A time series of structure factor amplitudes, IF, , is obtained, where the measured amplitudes correspond to a vectorial sum of structure factors of all intermediate states, with time-dependent fractional occupancies of these states as coefficients in the summation. Difference electron densities are typically obtained from the time series of structure factor amplitudes using the difference Fourier approximation (Henderson and Moffatt 1971). Difference maps are correct representations of the electron density distribution. The linear relation to concentration of states is restored in these maps. To calculate difference maps, a data set is also collected in the dark as a reference. Structure factor amplitudes from the dark data set, IFqI, are subtracted from those of the time-dependent data sets, IF,I, to get difference structure factor amplitudes, AF,. Using phases from the known, precise reference model (i.e., the structure in the absence of the photoreaction, which may be determined from... 

Chaotic behavior and synchronization in heterogeneous catalysis are closely related. Partial synchronization can lead to a complex time series, generated by superposition of several periodic oscillators, and can in some cases result in deterministically chaotic behavior. In addition to the fact that macroscopically observable oscillations exist (which demonstrates that synchronization occurs in these systems), a number of experiments show the influence of a synchronizing force on all the hierarchical levels mentioned earlier. Sheintuch (294) analyzed on a general level the problem of communication between two cells. He concluded that if the gas-phase concentration is the autocatalytic variable, then synchronization is attained in all cases. However, if the gas-phase concentration were the nonautocatalytic variable, then this would lead to symmetry breaking and the formation of spatial structures. When surface variables are the model variables, the existence of synchrony is dependent upon the size scale. Only two-variable models were analyzed, and no such strict analysis has been provided for models with two or more surface concentrations, mass balances, or heat balances. There are, however, several studies that focused on a certain system and a certain synchronization mechanism. 

Assuming that we have measured a series of concentrations over time/ we can define a model structure and obtain initial estimates of the model parameters. The objective is to determine an estimate of the parameters (CLe, Vd) such that the differences between the observed and predicted concentrations are comparatively small. Three of the most commonly used criteria for obtaining a best fit of the model to the data are ordinary least squares (OLS)/ weighted least squares (WLS)/ and extended least squares (ELS) ELS is a maximum likelihood procedure. These criteria are achieved by minimizing the following quantities/... 

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