Approximating a multivariable plane using a set of single variable equations

Question

I am attempting to make a multivariable equation that describes pump flow when 2 different characteristics are manipulated.

My equations are:

y = 8.1429x + 0.0057 when z=10

y = 7.3571x + 0.0043 when z=20

y = 6.5x + 0.005 when z=30

y = 5.7143x + 0.0036 when z=40

y = 4.9286x + 0.0021 when z=50

y = 4.1429x + 0.0007 when z=60

y = 3.2857x + 0.0014 when z=70

y = 2.5x + 9E-17 when z=80

y = 1.6429x + 0.0007 when z=90

y = x - 0.01 when z=100

y is pump flow

x is stroke length

z is stroke frequency

Any help is greatly appreciated,

Harrison

Answer 1

I don't know if this will help or not, but I made the slope and y intercept of each of your linear equations functions, f(z) and g(z)

y = f(z)x + g(z)

I then did a least squares line fit for both f(z) and g(z) to fit the slopes and y-intercepts.

f(z) = az +b

g(z) = cz + d

y = f(z)x + g(z)

f(z) = -0.08z + 8.932

g(z) = -0.0000718z + 0.0062

This is just an approximation. Might use a cubic spline fit for closer approximation instead of a line fit.

I hope this helps.

Answer 2

Of course with approximation there are going to be some choices you can make, but perhaps a natural starting point is a generalization of a least-squares method.

We can see from your equations that both the slope and the y-intercept change with z. That means you are looking for an equation

y = a(z)x + b(z)

where a(z) and b(z) are some functions of z. A natural starting point is to assume each of these functions is linear in z:

a(z) = a_1 z + a_0

b(z) = b_1 z + b_0

From here the pairs of z-values and the corresponding values of the coefficients (represented by a and b) give the data points needed to find a best choice for a_1, a_0, b_1, b_0.

See

https://www.wyzant.com/resources/files/729335/least_squares_for_multiple_variable_approximation

for description of how to take this data and set up an equation one can solve for the answer you want. The method used here requires some linear algebra knowledge which is not the only way to get to this answer, but is fairly standard and simple to reuse. From that link you can see a least-squares type approach gives

y = (-0.0802164z + 8.93334)x - 0.000120909z + 0.008

as a first attempt at approximating all of your lines with one function of x,y, and z. Whether this function is suitable for your needs will depend on what you need to do with it!

Further steps could involve things like using higher degree polynomials to give a(z) and b(z) in an effort to squeeze more accuracy out of the approximating function.