Modelling concerns the mathematical representation of the nature of the process with respect to its environment; the purpose of modelling and the type of data available are important considerations.

The following sections are included: introduction; smoothing methods of prediction; box and Jenkins method; other selected methods; and concluding remarks.

In this chapter, state-space modelling and optimal estimation of states using the Kalman filter were studied. Compared with the transfer function models based on input-output data, the state-space approach offers the additional flexibility of accommodating internal variables as states, which may not be accessible or measurable. The state-space approach permits use of a large number of widely studied and well-established methods and algorithms for estimation, prediction, smoothing and control. The Kalman filter can produce optimal state estimates under steady state conditions, even when the measurements are noisy, conditional on the noise being independent with Gaussian distribution.

Certain special features characterize a nonlinear process which can be represented by a single-stage or a multistage model, linear or nonlinear in the parameters.

Nonlinear series with or without periodicity as well as nonlinear input-output processes can be modelled using neural networks.

Predictive control aims at obtaining the predicted performance of the process as specified. This chapter describes some of the popularly used predictive control methods for linear systems. A real-life process is usually dynamic in nature, and it works in a stochastic environment; so it is necessary for the controller to be adaptive.

Some methods for separation or extraction of usable information from the available data have been presented. First, optimum smoothing in state-space framework was discussed, which was to familiarize the reader with the various issues connected with smoothing. This was followed by studies on bidirectional filtering, which can be used to perform smoothing with minimum lag or phase shift. The bidirectional processing is a characteristic feature which is incorporated in many algorithms for bias-free processing; for example the fixed interval smoother uses similar forward-backward passes, and the centred moving average used in the time series analysis is also effectively similar in concept. Orthogonal transformation offers a numerically robust method of smoothing and signal extraction. The smoothing is performed through the elimination of insignificant on singular value decomposition (SVD). Besides smoothing, the potential of the SVD based methods in signal extraction and pattern estimation in a noisy environment was also demonstrated through application studies. The approach depends on the repetitive nature of the signal component of interest, and hence the data are appropriately configured for analysis. A case study on fetal ECG extraction from maternal ECG showed that extraction is possible with only one signal (i.e. the maternal ECG signal from the abdominal lead), and irrespective of low signal to noise ratio; the other available methods of fetal ECG extraction require one or more additional signals. The application of orthogonal transformation for smoothing and filtering is an area of active research, and the present study has been only a glimpse of its enormous potential.

The trigonometric Fourier series representation of a periodic process f(t), with period T, was discussed in Sec.2.5.1. The derivation of exponential Fourier series representation is presented here. The sinusoidal functions can be expressed in terms of exponential functions as follows e = cosθ + lsinθ, l, iθ -iθΛ . 1, lθ -lθv . f-r cosθ = -(e + e ), smθ = -(e e ), i v-1. So the trigonometric Fourier series (2.5.2) can be expressed as f(t) = aο i ΣWeιηω'Ve-ιηωo1) ibηfeιηωoι-e“1"^)).

The two basic purposes of Centred Moving Averaging (CMA) are (i) estimation of trend in a time series without any time-lag, and (ii) reduction of the effects of random or spurious noise associated with the data. The general consequence of averaging is low-pass filtering, that is the high frequency components are attenuated whereas the low frequency components are retained. Any averaging which uses present and the past data only will produce a time-lag in the averaged data. In CMA, since both past and post data are used, the lag-free estimate of the series is produced.

This section of the book deals with predictor algorithm formulated for multivariable stochastic process.

The supportive details of the example presented in the electrical power load problem and air traffic problem are given in this appendix.

The atmospheric concentration of carbon dioxide in parts per million for 22 consecutive years from 1959 to 1980 measured at the Mount Mauna Loa observatory in Hawaii is given in a table; the data are presented row-wise for each year (from January to December) starting from 1959.

In this paper the monthly figures on the number of people unemployed in Germany during the period 1948 to 1978 are given. The monthly data for each year are presented row-wise.

The count the number of spots on the sun's surface is of interest in Astronomy and Climatology for geophysical reasons; the series is also of interest to time series analysts for the time-varying nature of the series. The daily observations from more than 50 observatories are used to arrive at the relative values of the sunspot numbers; the yearly averaged values for the years 1700 to 1987 are presented here.

The chemical oxygen demand (COD) can be considered to be an index of water pollution in the sea. COD concentration is monitored at a number of stations in the Osaka bay along with water temperature, transparency and dissolved oxygen concentration. Altogether 84 sets of monthly data are available, corresponding to the years 1976 to 1983. The output variable is the COD concentration, which is related to the input variables: water temperature, water transparency and dissolved oxygen concentration. The observed values of filtered COD (i.e. COD values of sea water free from suspended materials) are also presented.

SVR spectrum is a method of determining the period length of periodic components present, if any, in any signal or data sequence; the periodic components need not be sinusoidal. The data <y(k)> are arranged into the consecutive rows of a matrix (A) which is singular value decomposed; the generic term Singular Value Ratio (SVR) spectrum stands for the spectrum of a function (usually squared ratio) of the most dominant and other singular values against varying row lengths of the data matrix.

One of the fundamental works in predictive control is by Smith (1957), who designed a controller essentially free from the effects of the time delay. A typical process shows an inherent time delay between the input u, and the process output y. Let the time delay be expressed as G_d, given by exp(-sτ), in continuous time, s being the Laplace operator. The control action can at best force the output to be equal to the set point in a time equal to the dead time of the process. Faster control performance is not possible. Under-estimation of the time delay and unnecessary control action can lead to instability.

Use of transmittance matrices offers a straightforward method for simplifying a polynomial matrix multiplication problem into an ordinary matrix multiplication problem. The transmittance matrix formulation can be very useful in per forming state estimation. This appendix discusses the vector formulation of the transmittance matrix and its application for state estimation.

A low-pass filter is a frequency domain filter, which allows the specified low-frequency part of the signal or data sequence to pass through, whereas the higher frequency components are attenuated. The objective is to separate from the data undesirable high frequency components, which may be due to external disturbances or noise. The filter may be characterized by the pass-band, the transition band, the stop-band and the gain or the pass-band magnitude. The smaller the transition region, the sharper is the separation between the frequency components passed and those attenuated.

This appendix discussed the composite maternal and fetal ECG during gestation period by downsampling the digitized data.