The following four regression functions can be fit to the data in a Stats, Categorized, or Custom ternary plot. Note that these functions are derived from the equations for the respective standard polynomial functions, using the restriction that the sum of the values of the component (X, Y, Z) variables for each case is equal to a constant (e.g., 1.0). For example, the simple first-degree model:

V = a + b1*X + b2*Y + b3*Z

with the constraint X+Y+Z=1, can be adjusted by multiplying the intercept coefficient a by 1=X+Y+Z:

V =

This expression can be simplified to:

V = (a+b1)*X + (a+b2)*Y + (a+b3)*Z

or: V = b1' *X + b2' *Y + b3' *Z

The available "canonical" form polynomial regression functions are listed below (note that in these equations, the parameters are simply referred to as b1, b2, etc., and not b1', b2', etc.):

Linear Smooth (1st degree polynomial): V = b1*X + b2*Y + b3*Z

Quadratic Smooth (2nd degree polynomial): V = b1*X + b2*Y + b3*Z+ b12*X*Y + b13*X*Z + b23*Y*Z

Full Cubic Smooth: V = b1*X + b2*Y + b3*Z + b12*X*Y + b13*X*Z + b23*Y*Z + b12*X*Y*(X-Y) + b13*X*Z*(X-Z) + b23*Y*Z*(Y-Z) + b123*X*Y*Z

Special Cubic Smooth: V = b1*X + b2*Y + b3*Z + b12*X*Y + b13*X*Z + b23*Y*Z + b123*X*Y*Z

For more information about ternary representation of data, see Ternary Plots.