Online data normal distribution

The error model used in minimization assumes that the residuals have zero mean value and are normally distributed. The latter assumption could be violated, since not all of the deviations between the model and the data are of a stochastic nature. Analytical techniques, particularly automation of off-line analysis, such as gas and liquid chromatography and development of online analytical techniques (UV, FTIR, flow and sequential injection analysis) suppress the random scattering in the data to a minimum, and beautiful experimental curves can be plotted. Still, a lot of deviations appear between experimental and predicted data. The main reason originates from systematic deviations, which are easily recognized by graphical consideration of data sets, e.g. plotting the residuals as a function of dependent or independent variables. 

If the online data of steam rates resembles the normal distribution, we can readily determine the mean or average, the distribution frequency, standard deviation, and lower and upper bounds of the steam rates. 

To understand the frequency of letdown flow at different rates, the online data in Figure 21.1 are represented by the flow ranges and the frequency of steam rates in terms of counts in each range as shown Figure 21.2. Note that the counts for all intervals add up to 360, which is equal to the total data points in Figure 21.1. The profile in the figure resembles a bell-shaped normal distribution. Assume the normal distribution fits the actual distribution. This assumption can allow us to readily determine HP letdown average, flow distribution and variance. The calculations of these parameter are explained in the steps that follow. 

There are cases when trustable online for an operating parameter are not available, but the minimum and maximum operating limits are known from experience. In this case, the sample data set of normal distribution can be generated via... 

FIGURE 21.7. Comparison of online data with normal distribution. 

If the abnormality is not caused by the bad data, the fundamental question then becomes Is it justified to use a normal distribution to model the acmal distribution In general, it is rare that normal distribution fits the online data perfectly. As long as the variation pattern mimics the beU curve of normal distribution, it is reasonable to use the normal distribution to model the variation of real data. Use of normal distribution enables us to determine the true benefit based on variance. 

Statistical methods (such as F-test, Student s t-test, and x -test) can be used to prove the justification of using normal distribution to model online data. Usually, F-test is applied first to determine if a data set follows the normal distribution. Student s t-test is applied to determine the means or average assuming a common variance between a small set of sample data points and a large data pool. On the other hand, / -test can be applied to determine the variance for a small set of sample data points by assuming a common means between a small set of sample data points and a large data pool. For readers who are interested in this topic, please read SchmuUer (2009). 

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