Time Series Modelling

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. 

Building a Time Series Model Using Pilot Plant Data... 

Figure 2. Experimental trial used to Identify transfer function. In this experiment, the reactant flow rate was deliberately varied and the reactant temperature measured on-line in the pilot plant. This allowed us to identify the proper time series model.

At Rohm and Haas a committee of several experts contributed to the successes described In this paper. Discussions with Prof. John MacGregor (HcHaster University), Jeff Nathanson, Tom Shannon and Tom Throne were especially Important. Special thanks are due to Chris Altomare, who always had the proper equipment and Instrximentatlon ready for the pilot plant trials. We also would like to acknowledge Prof. Don Watts (Queens University) for assisting with the time series modeling and Prof. 

An autoregressive time series model (16) seems to be less suitable for cumulative distribution data. This technique is primarily designed for finding trends and/or cycles for data recorded in a time sequence, under the null-hypothesis that the sequence has no effect. 

The first time-series model of McCollister and Wilson yields results of about 0.4 for the mean absolute error divided by the mean value for the days in 1972, on the basis of parameters derived before 1972. By comparison, a persistence model yields about 0.5 for the same parameter, and the Los Angeles apcd forecasts lie between the time-series results and the persistence-model results. The hourly oxidant time-series model yields errors from 0.3 to 0.45, whereas persistence yields errors of 0.4-0.5. For carbon monoxide, the hourly-model results lie between 0.15 and 0.5—consistently below the persistence results. 

A non-linear time series model transforms an observed signal x[t into a white noise process e[t, and may be written in discrete form [Priestley, 1988] as ... 

Most of the above purposes can be fulfilled with modern time series models but the environmental scientist also needs a guide to clear interpretation of the results of the applied mathematical methods. The practician should be given time series analytical methods, i.e. mathematical techniques or computer programs which are relatively simple to use, as a tool for his daily work. The interpretation of the computations must be easy. More complicated time series models are, therefore, not included. 

The aim of the following calculations is to reduce the expense of sampling and analysis and to discover valid time series models at critical dates which make it possible to decide whether the water from the reservoir in Weida can be used as human drinking water, or if other sources have to be included. 

Frequently, concentration variations in environmental matrices not only concern themselves with one-dimensional cases, e.g. the time series of one parameter as discussed before, but also with many parameters which change simultaneously. In environmental analysis in particular, such time or local changes of environmental contaminants are very relevant [GEISS and EINAX, 1992], Multivariate time series models are available,... 

ARIMA modeling in the analysis of water quality data is discussed in detail by ZET-TERQVIST [1989]. Similar models were applied to the analysis of acid rain data [TERRY et al., 1985] and to the analysis of air quality data [JOHNSON and WIJN-BERG, 1981]. In general, these models are very rarely used in environmental analysis. We want to demonstrate the power of this kind of time series model and will also supply candidates suitable for use of these models for evaluation of time series. 

Stochastic Models for the Disturbances The type of stochastic process disturbances N-t occurring in practice can usually be modelled quite conveniently by statistical time series models (Box and Jenkins (k)). These models are once again simple linear difference equation models in which the input is a sequence of uncorrelated random Normal deviates (a. ) (a white noise sequence)... 

Comparing this with equation (3) shows that this can be considered as the output of a first order transfer function in response to a random input sequence. More generally, most stochastic disturbances can be modelled by a general autoregressive-integrated moving-average (ARIMA) time series model of order (p,d,q), that is,... 

Hipel, K.W. and McLeod, A.I. (1994) Time Series Modelling of Environmental and Water Resources Systems. Elsevier, Amsterdam. 

The effects of autocorrelation on monitoring charts have also been reported by other researchers for Shewhart [186] and CUSUM [343, 6] charts. Modification of the control limits of monitoring charts by assuming that the process can be represented by an autoregressive time series model (see Section 4.4 for terminology) of order 1 or 2, and use of recursive Kalman filter techniques for eliminating autocorrelation from process data have also been proposed... 

Autocorrelation in data affects the accuracy of the charts developed based on the iid assumption. One way to reduce the impact of autocorrelation is to estimate the value of the observation from a model and compute the error between the measured and estimated values. The errors, also called residuals, are assumed to have a Normal distribution with zero mean. Consequently regular SPM charts such as Shewhart or CUSUM charts could be used on the residuals to monitor process behavior. This method relies on the existence of a process model that can predict the observations at each sampling time. Various techniques for empirical model development are presented in Chapter 4. The most popular modeling technique for SPM has been time series models [1, 202] outlined in Section 4.4, because they have been used extensively in the statistics community, but in reality any dynamic model could be used to estimate the observations. If a good process model is available, the prediction errors (residual) e k) = y k)—y k) can be used to monitor the process status. If the model provides accurate predictions, the residuals have a Normal distribution and are independently distributed with mean zero and constant variance (equal to the prediction error variance). 

An alternative SPM framework for autocorrelated data is developed by monitoring variations in time series model parameters that are updated at each new measurement instant. Parameter change detection with recursive weighted least squares was used to detect changes in the parameters and the order of a time series model that describes stock prices in financial markets [263]. Here, the recursive least squares is extended with adaptive forgetting. 

Various multivariate regression techniques are outlined in Section 4.1. Section 4.2 introduces PCA-based regression and its extension to capture d3mamic variations in data. PLS regression is discussed in Section 4.3. Input-output modeling of d3mamic processes with time series models is introduced in Section 4.4 and state-space modeling techniques are presented in Section 4.5. 

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

Model identification is an iterative process. There are several software packages with modules that automate time series model development. When a model is developed to describe data that have stochastic variations, one has to be cautious about the degree of fit. By increasing model complexity (adding extra terms) a better fit can be obtained. But, the model may describe part of the stochastic variation in that particular data which will not occur identically in other data sets. Consequently, although the fit to the training data may be improved, the prediction errors may get worse. 

The d3Tiamic response of e k) can be expressed as an autoregressive moving average (ARMA) model or a moving average (MA) time series model ... 

Time series models of the output error such as Eq. 9.5 can be used to identify the dynamic response characteristics of e k) [148]. Dynamic response characteristics such as overshoot, settling time and cycling can be extracted from the pulse response of the fitted time series model. The pulse response of the estimated e k) can be compared to the pulse response of the desired response specification to determine if the output error characteristics are acceptable [148]. 

Since yMst is a random variable, SPM tools can be used to detect statistically significant changes. histXk) is highly autocorrelated. Use of traditional SPM charts for autocorrelated variables may yield erroneous results. An alternative SPM method for autocorrelated data is based on the development of a time series model, generation of the residuals between the values predicted by the model and the measured values, and monitoring of the residuals [1]. The residuals should be approximately normally and independently distributed with zero-mean and constant-variance if the time series model provides an accurate description of process behavior. Therefore, popular univariate SPM charts (such as x-chart, CUSUM, and EWMA charts) are applicable to the residuals. Residuals-based SPM is used to monitor lhist k). An AR model is used for representing st k) ... 

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