Draper point

In Fig. 5.24, MXs is reproduced with a logarithmic scale. The region of visible light is also indicated. It is only at sufficiently high temperatures that a significant portion of the emissive power is emitted within this wavelength interval. First, at the so-called Draper point at 798 K (525 °C), [5.9], will a heated body in dark... 

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. 

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. 

The well-known Box-Wilson optimization method (Box and Wilson [1951] Box [1954, 1957] Box and Draper [1969]) is based on a linear model (Fig. 5.6). For a selected start hyperplane, in the given case an area A0(xi,x2), described by a polynomial of first order, with the starting point yb, the gradient grad[y0] is estimated. Then one moves to the next area in direction of the steepest ascent (the gradient) by a step width of h, in general... 

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... 

In the early stages of much research, it is not always known which of the system inputs actually affect the responses from the system that is, it is not always known which inputs are factors, and which are not. One point of view describes all Inputs as factors, and then seeks to discover which are significant factors, and which are not significant. How would you design an experiment (or set of experiments) to prove that a factor exerted a significant effect on a response How would you design an experiment (or set of experiments) to prove that a factor had absolutely no effect on the response [See, for example, Fisher (1971), or Draper and Smith (1981).]... 

The choice of the number of center points and the blocking of composite designs is discussed in Myers [11], Box and Draper [12], and Khuri and Cornell [13]. 

The arrangement of the points in the concentration triangle is shown in Fig. 3.26. Draper and Lawrence suggested for second-order polynomials of Eq. (3.125) as applied to ternary systems, the designs containing from 8 to 15 design points. Parameters for the Draper-Lawrence designs (in fractions of m) at q=3, ni=2 and n2=3, are summarized in Table 3.46. 

Parameters (in fractions of m) for some designs of Draper-Lawrence containing no more than 12 points, at q=4, nj=l, n2=2, are provided in Table 3.48. 

A Draper-Lawrence design containing 13 points is performed (Table 3.51). It is convenient to treat the subregion studied as a concentration triangle in the new coordinate system (X], X2, X3 ) ... 

Geometric interpretation of design points in Draper-Lawrence design is shown in Fig. 3.29. 

We can digress at this point to consider two laws of photochemistry thal feature in most introductions to the subject. First, the Grotthuss-Draper law states, in essence, that only light absorbed by a molecule can be effective in bringing about chemical change. This may seem obvious when considered at the molecular level—if the photon energy is not made available to the molecule by absorption, an electronically excited state cannot be produced, and no photochemical change can result. The law s importance is in its practical... 

Face-centered Draper-Lin small composite design with four central points... 

The above methods are known, but under-utilized in practice. More on this subject, from a statisticians point of view, can be found in a book by Draper and Smith (1981). 

These are almost the same values as in the example 6.3.2 in (Box Draper, 2007) though we have used a different model with significant interaction terms included. The reason for this is that the starting point is the centre point where the interaction terms vanish because the coefficients are multiplied by zeros. 

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