Rotatable design

The use of rotatable designs usually makes sense only in normalized factor spaces (each factor divided by d ) because it is difficult to define distance if the factors are measured in different units. For example, if x, is measured in °C and is measured 

However, it is not possible to add °C and min In a normalized factor space the factors are unitless and there is no difficulty with calculating distances. Coded rotatable designs do produce contours of constant response in the uncoded factor space, but in the uncoded factor space the contours are usually elliptical, not circular. 

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Describe (where possible) the production system components (e.g. crop or livestock health management practices, soil fertilisation methods, crop rotation designs, livestock feeding and husbandry regimes, crop varieties/livestock breeds used) responsible for differences in food quality and safety between production systems ... 

Upon integration and rearrangement the screw rotation design Eq. A5.13 is obtained. 

Assuming the same s] as was used to draw Figure 12.24, the uncertainty surface for the rotatable design of Equation 12.81 is shown in Figure 12.26. The uncertainty depends only on the distance from the center of the design i.e., the contours of constant uncertainty are circular about the center of the design. 

Figure 12.26 Standard uncertainty for estimating one new response as a function of the factors a , and Xj for a rotatable design. Compare with Figure 12.24. See text for discussion.

The effects of visual scaling and numerical coding are very similar. However, the distortions caused by numerical coding are not always as readily apparent as some distortions caused by visual scaling. For example, rotatable designs in coded factor spaces might not produce rotatable designs in uncoded factor spaces (see Section 12.9). We simply warn the reader that concepts such as rotatable , circular . 

The normalized information at the center (and at the edges) of the factor space in Figure 13.3 is less than the normalized information at the center (and at the edges) in Figure 13.2. These effects are a result of the relative compactness of the star points in this rotatable design which allows the FSOP model to flex more at the comers of the factor space and, consequently, at the center as well. 

Figure 13.15 A pentagonal rotatable design with center point. = 0, = 3.

Box, G.E.P., and Behnken, D.W. (1960a), Simplex-Sum Designs A Class of Second Order Rotatable Designs Derivable from those of First Order, Ann. Math. Statist., 31, 838-864. 

The industrial use of twin-screw extruders for this purpose revolves extensively, but not exclusively, around intermeshing co-rotating variants. Closely in-termeshing counter-rotating designs are widely used for profile extrusion of UPVC dry-blends since they permit close temperature control and exhibit a high conveying efficiency due to the positive displacement of material where the screws intermesh [150]. 

There are a number of other designs that have advantages in particular circumstances. Many are just variants on the theme of maximally spanning the factor space. The Box-Behnken design is a two-level, spherical, rotatable design (Box and Behnken 1960). For three factors it has experiments at the center and middle of each edge of a cube (figure 3.14). 

In this example, orthogonality of all factor effects has been achieved by including additional center points in the coded rotatable design of Equation 11.81. Orthogonality of some experimental designs may be achieved simply by appropriate coding (compare Equation 11.26 with Equation 11.20, for example). Because orthogonality is almost always achieved only in coded factor spaces, transformation of... 

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