Factors aliasing

Fixed Effects, Random Effects, Main Effects, and Interactions Nested and Crossed Factors Aliasing and Confounding... 

A second major source of computational difficulties associated with uniform prior-prejudice distributions is connected with the extremely fine sampling grids that are needed to avoid aliasing effects in the numerical Fourier synthesis of the modulating factor in (8). To predict the dependence of aliasing effects upon the prior prejudice, we need to examine more closely the way the MaxEnt distribution of scatterers is actually synthesised from the values of the Lagrange multipliers X. 

BUSTER chooses the minimal grid necessary to avoid aliasing effects, based on the prior prejudice used and on the fall-off of the structure factor amplitudes with resolution for the 23 K L-alanine valence density reconstruction the grid was (64 144 64). The cell parameters for the crystal are a = 5.928(1)A b = 12.260(2)A c = 5.794(1) A [45], so that the grid step was shorter than 0.095 A along each axis. 

The calculation of the thermally-smeared core fragment and the valence monopoles densities was carried out by a Fourier transform of a set of aliased structure factors computed with the program VALRAY [46] details of this calculation have been published elsewhere [49],... 

Fractional replication causes confusion among the factor effects. This confusion is called confounding or aliasing . To see this, compare the signs in the columns in the fractional factorial design ... 

Suppose that the experimenter runs the 2 fractional factorial design shown in Table 2.3. With this design each main effect is aliased with the two-factor interaction composed of the other two factors that is, is aliased with x x, x is aliased with and x is aliased with x x. This can be verified by multiplying together the appropriate columns, as was done for x Xy... 

Some protection against the effect of biases in the estimation of the first-order coefficients can be obtained by running a resolution IV fractional factorial design. With such a design the two-factor interactions are aliased with other two-factor interactions and so would not bias the estimation of the first-order coefficients. In fact the main effects are aliased with three-factor interactions in a resolution IV design and so the first-order effects would be biased if there were third-order coefficients of the form xxx, in... 

The relationship D = ABC in this design is called the generator. The factor D and the three-factor interaction ABC are called aliases of one another because they are confounded. All aliases can be determined with the help of the defining relation or defining contrast (I). It is obtained by multiplying the effects occuring in the generator. 

The alias of any effect can be obtained by multiplying the effect with the defining relation, with as an additional rule that when a term appears an even number of times this term disappears from the product. For instance the aliases of factor A and of interaction AB are respectively ... 

If there were no noise in the world, these factors would control the required sampling rate. However, because noise exists and noise can be aliased as well as data, the maximum noise frequency passed by the electronics system actually determines the required sampling rate. 

Under less restrictive noise/bandwidth considerations, one might drop the density to 6 points per resolution element at the risk of some minor noise aliasing. However, deconvolution by a factor of 3 would leave only two points per FWHM of a Gaussian spectrum-a number sufficient to characterize the spectrum, although display and measurement are difficult. 

Bracketing and matrixing are the two reduced designs recommended by the FDA [9]. Each of these methods applies to different situations. Using both of them simultaneously may reduce the ability of the study to determine the shelf life since factor combinations can be confounded due to the aliasing effect [10]. 

Table 3.4. Fractionating a two-way, four-factor design by aliasing the three-way 1x2x3 effect with the main effect of 4.

The given ratio helps to determine aliased/confounded effects. For this, it is necessary to multiply successively both sides of the defining contrast by factors from matrix columns. Factor X4 is in this case obtained ... 

It is then multiplied by each factor from the design 23"1. If the given yields offer the square of the factor, it is automatically replaced by the number one. Aliased/confounded effects for the observed half-replica are given by these ratios ... 

Thus, in this 24 1 fractional factorial design, each factorial effect has a single alias the four main effects are aliased with the four three-factor interactions, and the two-factor interactions are aliased with each other in pairs. Only one effect in each alias chain can be estimated. Aliases can be found by multiplying the effect of interest through the defining relation for the design. For example, the alias of A is found by multiplying A by I = ABCD, which produces A = A2 = BCD = BCD, because A2 is the identity column. 

The 27-4 design is a saturated design by this we mean that the number of factors, / = 7, is one less than the number of runs, n = <8. The alias relationships here are somewhat more complicated. Specifically, if we ignore all interactions of order three or higher, each main effect is aliased with a chain of three two-factor interactions ... 

A resolution III design has at least some main effects aliased with two-factor interactions so the27-3 design in Table 2 is a resolution III design (often denoted... 

A resolution IV design has main effects clear of (not aliased with) two-factor interactions, but at least some two-factor interactions are aliased with each other so the 24 1 design in Table 1 is a resolution IV design (often denoted 24IV]). 

Note that this choice of generators results in 15 of the 21 two-factor interactions being aliased with each other across seven alias chains. However, if instead we choose F = ABC D and G = ABDE as the generators, then another resolution IV design results, but in this design the two-factor interactions are aliased with each other as follows. 

Figure 3 shows a half-normal probability plot of the effect estimates obtained from the Design-Expert software package. Three main effects, A (pressure), B (power), and E (gap) are important. Because the main effects are aliased with three-factor interactions, this interpretation is probably correct. There are also two two-factor interaction alias chains that are important, AB = CE and AC = BE. Because AB is the interaction of two strong main effects, those of pressure and... 

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