Time series modeling examples

The precision of time series predictions far into the future may be limited. Time series analysis requires a relatively large amount of data. Precautions are necessary if the time intervals are not approximately equal (9). However, when enough data can be collected, for example, by an automated process, then time series techniques offer several distinct advantages over more traditional statistical techniques. Time series techniques are flexible, predictive, and able to accommodate historical data. Time series models converge quickly and require few assumptions about the data. 

Sources of disturbances considered in this example are categorized in three classes. First, the production plants are stochastic transformers, i.e. the transformation processes are modelled by stationary time series models with normally distributed errors. The plants states are modelled by Markov models as introduced before. The corresponding transition matrices are provided in the appendix in Table A.15 and Table A.16. Additionally, normally distributed errors are added to simulate the inflovj rates with e N (O, ) where oj is the current state of the plant. 

If the empirical dynamic model is hnear, one could use for example a time series model. Different model types are available and will be discussed in a subsequent chapter. 

The first level of complexity corresponds to simple, low uncertainty systems, where the issue to be solved has limited scope. Single perspective and simple models would be sufficient to warrant with satisfactory descriptions of the system. Regarding water scarcity, this level corresponds, for example, to the description of precipitation using a time-series analysis or a numerical mathematical model to analyze water consumption evolution. In these cases, the information arising from the analysis may be used for more wide-reaching purposes beyond the scope of the particular researcher. 

The limits for part (b) are at the endpoints of a vertical line in Figure 15.14 that corresponds to the residence time distribution for two tanks in series. The maximum mixedness point on this line is 0.287 as calculated in Example 15.14. The complete segregation limit is 0.233 as calculated from Equation (15.48) using/(/) for the tanks-in-series model with N=2 ... 

Very rarely are measurements themselves of much use or of great interest. The statement "the absorption of the solution increased from 0.6 to 0.9 in ten minutes", is of much less use than the statement, "the reaction has a half-life of 900 sec". The goal of model-based analysis methods presented in this chapter is to facilitate the above translation from original data to useful chemical information. The result of a model-based analysis is a set of values for the parameters that quantitatively describe the measurement, ideally within the limits of experimental noise. The most important prerequisite is the model, the physical-chemical, or other, description of the process under investigation. An example helps clarify the statement. The measurement is a series of absorption spectra of a reaction solution the spectra are recorded as a function of time. The model is a second order reaction A+B->C. The parameter of interest is the rate constant of the reaction. 

Using the nitrate time series example, the effect of a trend model with additive seasonality is shown in Fig. 6-7. 

Tab. 6-1. Seasonal differences in an additive model from the nitrate time series example...

Application of the ARIMA Modeling to the Example Time Series... 

Stochastic modeling is used when a measurable output is available but the inputs or causes are unknown or cannot be described in a simple fashion. The black-box approach is used. The model is determined from past input and output data. An example is the description of incomplete mixing in a stirred tank reactor, which is done in terms of contributions of dead zones and short circuiting. In these cases, a sequence of output called a time series is known, but the inputs or causes are numerous and not known in addition, they may be unobservable. Though the causes for the response of the system are unknown, the development of a model is important to gain understanding of the process, which may be used for future planning. 

It may appear that Table 1 contains an essentially complete summary of patterns that may form in electrochemical systems. This impression is misleading, since Table 1 only roughly summarizes results observed so far or predicted with models. These are investigations concentrating on phenomena that can be described with two essential variables (two-component systems). This survey is certainly not yet completed. Furthermore, numerous examples of current or potential oscillations involve complex time series. Only in a few cases does the complex time series result from the spatial patterns. In most cases, the additional degree of freedom will be from a third dependent variable, such as from a concentration that adds an additional feedback loop into the system, as discussed in Section 3.1.3. Spatial pattern formation in three-variable systems is an area that currently develops strongly in nonlinear dynamics. In the electrochemical context, the problem of pattern formation in three-variable systems has not yet been approached. 

A key factor in modeling is parameter estimation. One usually needs to fit the established model to experimental data in order to estimate the parameters of the model both for simulation and control. However, a task so common in a classical system is quite difficult in a chaotic one. The sensitivity of the system s behavior to the initial conditions and the control parameters makes it very hard to assess the parameters using tools such as least squares fitting. However, efforts have been made to deal with this problem [38]. For nonlinear data analysis, a combination of statistical and mathematical tests on the data to discern inner relationships among the data points (determinism vs. randomness), periodicity, quasiperiodicity, and chaos are used. These tests are in fact nonparametric indices. They do not reveal functional relationships, but rather directly calculate process features from time-series records. For example, the calculation of the dimensionality of a time series, which results from the phase space reconstruction procedure, as well as the Lyapunov exponent are such nonparametric indices. Some others are also commonly used ... 

Type of endpoint. The type of endpoint recording is essential for the application of different types of mixture toxicity models. Endpoints measured at only 1 point in time may only be used to derive concentration-response-related parameters, such as ECx or LCx or NOECs. Continuous recording or at least repeated recording of responses may allow for time series analysis. Time-related responses may, for example, be used for the derivation of kinetic parameters by applying pharmacokinetic/dynamic models (like the PBPK models in human toxicology see, e.g., Krishnan et al. 1994) or... 

In some cases cyclic events occur, dependent, for example, on time of day, season of tire year or temperature fluctuations. These can be modelled using sine functions, and are tire basis of time series analysis (Section 3.4). In addition, cyclicity is also observed in Fourier spectroscopy, and Fourier transform techniques (Section 3.5) may on occasions be combined with methods for time series analysis. 

Figure 10.7 shows examples of time series of the area average over Europe of the GRG species for model level 55 (PEL niveau) of the three IFS runs and the MOZART simulation. If total CTM tendencies (IFS tend) are applied, the IFS can imitate the CTM up to a forecast length of 48 h. Differences are obvious if the IFS vertical transport scheme is apphed, because the vertical transport schemes... 

Fig. 1. Some examples of heterogeneous oscillations observed experimentally, (a) Sinusoidal time series for H2/O2 on a Pt wire (from Ref. 41). (b) Relaxation oscillations for CO/O2 on Pt/Al203 (from Ref. 100). (c) Oscillations after a single period doubling for CO/O2 on Pt(l 10) (from Ref. 231). (d) Model and experimental (inset) quasiperiodic oscillations for NO/CO on supported Pd (from Ref. 232). (e) Deterministic chaos produced by a period doubling sequence for CO/O2 on Pt(IlO) (from Ref. 231).

The multifractal behavior of time series such as SRV, HRV, and BRV can be modeled using a number of different formalisms. For example, a random walk in which a multiplicative coefficient in the random walk is itself made random becomes a multifractal process [59,60], This approach was developed long before the identification of fractals and multifractals and may be found in Feller s book [61] under the heading of subordination processes. The multifractal random walks have been used to model various physiological phenomena. A third method, one that involves an integral kernel with a random parameter, was used to model turbulent fluid flow [62], Here we adopt a version of the integral kernel, but one adapted to time rather than space series. The latter procedure is developed in Section IV after the introduction and discussion of fractional derivatives and integrals. 

Principal components analysis has also been applied to array time series data (83-85) and a limited number of principal components usually accounts for the essential features of the data set, allowing considerably reduced complexity for example, the sporulation data was modeled using as few as two principal components (83). 

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