What is a pseudo-component?

Pseudo-components are coded variables used to simplify design construction and model fitting, and reduce the correlation between component bounds in constrained designs. Constrained designs (those in which you specify lower and/or upper bounds) produce coefficients which are highly correlated.
  • Lower bounds are necessary when any of the components must occur in the mixture. For example, lemonade must contain lemon juice.
  • Upper bounds are necessary when the mixture cannot contain more than a specified proportion of an ingredient. For example, a cake mixture cannot contain more than 5% baking powder.

Usually, you can reduce the correlations between the coefficients by transforming the components to pseudo-components. Pseudo-components, in effect, rescale the constrained data area so the minimum amount allowed (the lower bound) of each component is zero.

The following table shows two components expressed in amounts, proportions, and pseudo-components. Suppose the total mixture is 50 ml. Let X1 and X2 be the amount scale. Thus X1 + X2 = 50. Suppose X1 has a lower bound of 20 (this means that the upper bound of X2 is 50 minus 20, or 30). The following are some points on the three scales:

Amounts Proportions Pseudo-components
X1 X2 X1 X2 X1 X2
50 0 1.0 0.0 1.0 0.0
20 30 0.4 0.6 0.0 1.0
35 15 0.7 0.3 0.5 0.5
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