Autocorrelated data

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. 

A principle H = maximum was first proposed by J. P. Burg (1967) in the estimation of power spectra from autocorrelation data. There is one condition under which estimator (37) goes into the Burg form, described next. 

It is important to have reliable laser diagnostics, preferably on a shot-to-shot basis this is possible for a 10 Hz system. To characterise the temporal profile of the pulses, single-shot spectra and autocorrelation data can establish whether the laser pulses are Fourier-transform limited. For lasers with a sech profile, a product AvAtxO.32 should be achieved. It is also important to monitor the focal spot, its Airy disc and average intensity. Alternatively, a reasonable measure of the focused intensity can be obtained using Xe gas and the known threshold intensities for producing the various stages of ionization [12]. 

Some statistics concepts such as mean, range, and variance, test of hypothesis, and Type I and Type II errors are introduced in Section 2.1. Various univariate SPM techniques are presented in Section 2.2. The critical assumptions in these techniques include independence and identical distribution [iid) of data. The independence assumption is violated if data are autocorrelated. Section 2.3 illustrates the pitfalls of using such SPM techniques with strongly autocorrelated data and outlines SPM techniques for autocorrelated data. Section 2.4 presents the shortcomings of using univariate SPM techniques for multivariate data. 

Two alternative methods for monitoring processes with autocorrelated data are discussed in the following sections. One method relies on the existence of a process model that can predict the observations and computes... 

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. 

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) ... 

FW Faltin and WH Woodall. Some statistical process control method-s for autocorrelated data - discussion. J. Quality Technology, 23 194-197, 1991. 

Montgomery, D. C., and Mastrangelo, C. M. (1991), Some Statistical Process Control Methods for Autocorrelated Data, Journal of Quality Technology, Vol. 23, No. 3, pp. 179-193. 

Figure 5. Sublayer burst period data in Newtonian and drag-reducing flows. The solid line represents both visual and hot-wire data in a water channel flow. Open and closed circles in the upper part of the plot represent two-dimensional visual water and polymer data, with the dashed line a pressure gradient correction. All other symbols are autocorrelation data, open for solvent, closed for polymer solutions (including open and closed circles in the lower part of the plot). Ekses are water, circled ekses are polymer circled dots are water, half-black squares and circles are polymer.

Autocorrelation data were analyzed using the ciimulanl method (see subsection 3.3.2). Sec Figure 3.60 as an example. By fitting two cumulants to experimental data, F/2q = 2.38 10 ciiY/s was found. The results obtained are recorded in Table 3.4. 

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