Adding center points

When an unreplicated experiment is run, the error or residual sum of squares is composed of both experimental error and lack-of-fit of the model. Thus, formal statistical significance testing of the factor effects can lead to erroneous conclusions if there is lack-of-fit of the model. Therefore, it is recommended that the experiment be replicated so that an independent estimate of the experimental error can be calculated and both lack-of-fit and the statistical significance of the factor effects can be formally tested. 

In some experimental contexts, however, each experimental run is expensive. Thus it is infeasible to replicate each design point of the experiment to obtain an estimate of the experimental error. 

When all of the variables are quantitative, an estimate of the experimental error can be obtained by adding to the full factorial, fractional factorial or Plackett-Burman design, a number of runs at the center of the design. The center of the design is the midpoint between the low and high settings of the two-level factors in the experiment. Thus, if there are p variables, and the levels of the variables have been coded (-1, +1), then the center of the design is (Xj, = (0, 0,. .., 0). If the 

Another reason for augmenting the two-level design with center points is that these points allow for an overall test of curvature. It is clear that with only two levels for each variable it is impossible to detect any quadratic effect of the variables. Thus, the underlying model is assumed to 

If the F-test is significant then there is evidence of a quadratic effect due to at least one of the variables. With the present design, however, the investigator will not be able to determine which of the variables has a quadratic effect on the response. Additional experimentation, perhaps by augmenting the current design with some star points to construct a central composite design (see section on central composite designs below), will need to be conducted to fully explore the nature of the quadratic response surface. 

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If the true model contains quadratic terms then the estimate of the intercept, Pq, of the first-order model will be biased. The lack-of-fit of the first-order model due to quadratic effects can be tested by adding center points to the design. 

To the design points obtained are added center points (centroids) of two-, three-,. .., and (q-l)-dimensional faces of the polyhedron and its center point. Coordinates of a central point are determined by taking average coordinates of previously chosen vertices ... 

In this model, the regression coefficients of the pure quadratic terms (the fijj) are not estimable because the typical screening design has all factors at only two levels. However, the experimenter should be alert to the possibility that the second-order model is required. By adding center points to the basic 2f factorial design we can obtain a formal test for second-order curvature, that is, a test of the null hypothesis... 

Adding center points (experiments at the center of the domain, coded co-ordinates 0, 0,... 0 is useful for factorial and screening experiments), even though they do not enter into the calculation of the model equation because ... 

In such small designs it is advisable to carry out all 8 experiments, adding perhaps as the 9th experiment an additional center point condition. This... 

A two levels of full factorial experimental design with three independent variables were generated with one center point, which was repeated[3]. In this design, F/P molar ratio, Oh/P wt%, and reaction temperature were defined as independent variables, all receiving two values, a high and a low value. A cube like model was formed, with eight comers. One center point (repeated twice) was added to improve accuracy of the design. Every analysis results were treated as a dependent result in the statistical study. 

Three-dimensional electron densities have no boundaries they converge to zero exponentially with distance from the nuclei of the peripheral atoms in the molecule. Considering a single, isolated molecule, the exact quantum-mechanical electron density becomes zero in a strict sense only at infinite distance from the center of mass of the molecule. Consequently, the electron density is not a compact set, just as the embedding three-dimensional Euclidean space E3 is not compact either. However, the three-dimensional Euclidean space E3, as a subset of a four-dimensional Euclidean space E4, can be slightly extended (for example, by adding one point) and made compact by various compactification techniques. 

An example is the relatively simple moving average filter. In case of a digitized signal, the values of a fixed (odd) number of data points (a window) are added and divided by the number of points. The result is a new value of the center point. Then the window shifts one point and the procedure, which can be considered as a convolution of the sipal with a rectangular pulse function, repeats. Of course, other functions like a triangle, an exponential and a Gaussian, can be used. 

Finally, the problem was resolved by irradiating standards and mixtures of standards in a factorial experiment. The experiment design was a full factorial experiment with three variables, mercury, selenium, and ytterbium, at two levels with replication and with a center point added to test higher order effects. The pertinent information on treatments and levels of variables are shown in Table VII. 

A five-level-five-factor CCRD was employed in this study, requiring 32 experiments (Cochran and Cox, 1992). The fractional factorial design consisted of 16 factorial points, 10 axial points (two axial points on the axis of each design variable at a distance of 2 from the design center), and 6 center points. The variables and their levels selected for the study of biodiesel synthesis were reaction time (4-20 h) temperature (25-65 °C) enzyme amount (10%-50% weight of canola oil, 0.1-0.5g) substrate molar ratio (2 1—5 1 methanol canola oil) and amount of added water (0-20%, by weight of canola oil). Table 9.5 shows the independent factors (X,), levels and experimental design coded and uncoded. Thirty-two runs were performed in a totally random order. 

The three-point interval search starts by evaluating f x) at the upper and lower bounds, X] and xu, and at the center point (x + Xu)l2. Two new points are then added in the midpoints between the bounds and the center point, at (3xl + Xu)/4 and (xl + 3xu)/4, as shown in Figure 1.11. The three adjacent points with the lowest values of f x) (or the highest values for a maximization problem) are then used to define the next search range. 

A non-CCD is another alternative if the program is conducted sequentially. If the results from the 2 design indicate that the factor space should be extended in a particular direction, then two star points can be added to a comer as shown in Figure 2. This design also meets all the requirements for a second-order design. For four or more factors, a CCD might use an imbedded fractional factorial design plus center points and the 2k star points, where k is the number of factors. 

At this point, the required experiments can be defined. For this purpose, the levels (e.g., -a, -1,0, -f1, H-a) in the theoretical experimental design (e.g., Tables 2.8, 2.14, and 2.9) are replaced by the real factor levels (e.g., Tables 2.2-2.4, respectively).This results in the experimental conditions for each experiment. The dummy factor columns in PB designs can be ignored at this point. Often a number of replicated experiments at nominal or center point conditions are added to the setup (see above). 

One crystal system can have all four space lattices (P, I, F, and C), while some crystal systems can have only the P-lattice. For each crystal system, the I-, F-, or C-lattices have unit cells that contain more than one lattice point, since we have added lattice points in various centered positions. Cells with more than one lattice point are sometimes known as multiple-primitive unit cells. The same axes refer to all the space lattices in a given crystal system, so all of these conventional, centered unit cells will display exactly the same rotational symmetry as does the corresponding P-cell in a given crystal system. 

The assignation of axes of reference in relation to the rotational symmetry of the crystal systems defines six lattices that, by definition, are primitive or P-lattices. To determine if new lattices can be formed from these P-lattices, one must determine if more points can be added so that the lattice condition is still maintained, and whether this addition of points alters the crystal system. For example, if one starts with a simple cubic primitive lattice and adds other lattice points in such a way that a lattice still exists, it must happen that the resulting new lattice still possesses cubic symmetry. Since the lattice condition must be maintained when new points are added, the points must be added to hightly symmetric positions of the P-lattice. These types of positions are (a) a single point at the body center of each unit cell, (b) a point at the center of each independent face of the unit cell, (c) a point at the center of one face of the unit cell, and (d) the special centering positions in the trigonal system that give a rhombohedral lattice. 

The experimental design normally used to determine the values of the coefficients of the special cubic model is called the simplex centroid, which we obtain by simply adding a center point, corresponding to a 1 1 1 ternary mixture, (xi,X2,X3) = (5,5,5) to the simplex lattice design. The coefficient of the cubic term is given by... 

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