Draper Smith

Some nomenclature is in order here bo is called the constant or offset term or intercept b is the regression coefficient x is the independent or predictor variable and y is the dependent or response variable is the residual. More regression nomenclature can be found, e.g., in Draper Smith [Draper Smith 1998],... 

Values of y, are measured for a sufficient number of values of x, or of the x,s and then the sum of squares of the e,-s is minimized [Box Draper 1987, Draper Smith 1998], Regression equations such as Equations (3.23)—(3.25) assume that the xs are error-free and that the error is in y. Dealing with possible error in the xs is important and some of the regression techniques presented below can handle this problem. 

Statistical theory requires that a regression model has to be built from an overdetermined system. For this reason it is required that there should be at least 3 to 5 lack-of-fit degrees of freedom (niof) available as a check on the suitability of the model (Brereton, 1992 Draper Smith, 1981). Hence in this case, for the simplest linear regression model using a total number of n sample points of which there are n, replicates, the maximum required number of model degrees of freedom, df, excluding bias, becomes df = n -n,- niof - 1 = 420 - 402 - 3 -1 = 14. 

When there are sufficient data at different temperatures, the temperature dependence of the parameters is reflected in the confidence ellipses (Bryson and Ho, 1969 Draper and Smith,... 

Draper, N. R., Smith, H. "Applied Regression Analysis," John Wiley, New York (1966). 

In the maximum-likelihood method used here, the "true" value of each measured variable is also found in the course of parameter estimation. The differences between these "true" values and the corresponding experimentally measured values are the residuals (also called deviations). When there are many data points, the residuals can be analyzed by standard statistical methods (Draper and Smith, 1966). If, however, there are only a few data points, examination of the residuals for trends, when plotted versus other system variables, may provide valuable information. Often these plots can indicate at a glance excessive experimental error, systematic error, or "lack of fit." Data points which are obviously bad can also be readily detected. If the model is suitable and if there are no systematic errors, such a plot shows the residuals randomly distributed with zero means. This behavior is shown in Figure 3 for the ethyl-acetate-n-propanol data of Murti and Van Winkle (1958), fitted with the van Laar equation. 

In many process-design calculations it is not necessary to fit the data to within the experimental uncertainty. Here, economics dictates that a minimum number of adjustable parameters be fitted to scarce data with the best accuracy possible. This compromise between "goodness of fit" and number of parameters requires some method of discriminating between models. One way is to compare the uncertainties in the calculated parameters. An alternative method consists of examination of the residuals for trends and excessive errors when plotted versus other system variables (Draper and Smith, 1966). A more useful quantity for comparison is obtained from the sum of the weighted squared residuals given by Equation (1). 

Draper, N., Smith, HApplied Regression Analysis, 2nd edition, John Wiley and Sons, New York, 1981. 

Draper RD, Smith D. Applied regression analysis, New York, Wiley, 1987. 

Draper, N.R. and Smith, H., 1966, Applied Regression Analysis , Wiley, New York. 

Algebraic equations Steady state of CSTR with first-order kinetics. Algebraic solution and optimisation (least squares. Draper and Smith, 1981). Steady state of CSTR with complex kinetics. Numerical solution and optimisation (least squares or likelihood function). 

Linear models with respect to the parameters represent the simplest case of parameter estimation from a computational point of view because there is no need for iterative computations. Unfortunately, the majority of process models encountered in chemical engineering practice are nonlinear. Linear regression has received considerable attention due to its significance as a tool in a variety of disciplines. Hence, there is a plethora of books on the subject (e.g., Draper and Smith, 1998 Freund and Minton, 1979 Hocking, 1996 Montgomery and Peck, 1992 Seber, 1977). The majority of these books has been written by statisticians. 

Data on the thermal isomerization of bicyclo [2,1,1] hexane were measured by Srinivasan and Levi (1963). The data are given in Table 4.4. The following nonlinear model was proposed to describe the fraction of original material remaining (y) as a function of time (x,) and temperature (x2). The model was reproduced from Draper and Smith (1998)... 

Draper NR, Smith H (1981) Applied regression techniques, 2nd edn. Wiley, New York Edgeworth FY (1887) On observations relating to several quantities. Hermathena 6 279... 

Draper N. and Smith H., Applied Regression Analysis, Second Edition, John Wiley and Sons (1981). 

Regression Algorithms. The fitting of structural models to X-ray scattering data requires utilization of nonlinear regression techniques. The respective methods and their application are exhausted by Draper and Smith [270], Moreover, the treatment of nonlinear regression in the Numerical Recipes [154] is recommended. 

Draper and Smith [1] discuss the application of DW to the analysis of residuals from a calibration their discussion is based on the fundamental work of Durbin, et al in the references listed at the beginning of this chapter. While we cannot reproduce their entire discussion here, at the heart of it is the fact that there are many kinds of serial correlation, including linear, quadratic and higher order. As Draper and Smith show (on p. 64), the linear correlation between the residuals from the calibration data and the predicted values from that calibration model is zero. Therefore if the sample data is ordered according to the analyte values predicted from the calibration model, a statistically significant value of the Durbin-Watson statistic for the residuals in indicative of high-order serial correlation, that is nonlinearity. 

Draper and Smith point out that you need a minimum of fifteen samples in order to get meaningful results from the calculation of the Durbin-Watson statistic [1], Since the... 

The discussions of DW are on p. 69 and 181-192 of Draper and Smith (third edition -the second edition contains a similar but somewhat less extensive discussion). They also include an algorithm and tables of critical values for deciding whether the correlation is statistically significant or not. You might also want to check out page 64 for the proof that the linear correlation between residuals and predicted values from the calibration is zero. 

---