Experimental Design¶
Important goal: equalize leverage of every point during multiple regression $$ h_{ii} = \dfrac{k}{n}, \forall i $$
Known Functional Form¶
Non-dimensionalization
Actions¶
- Simplify differential equations
- Rescale variables to unitless form
- Get rid of unnecessary parameters
- Reduce number of experiments needed to test hypothesis
Rules¶
- Identify differential equation
- Identify independent & dependent vars
- Replace each of them with a quantity scaled relative to characteristic unit of measure to be determined
- Divided through by the coefficient of the highest order polynomial/derivative term
- Scale boundary conditions
- Choose the definition of the characteristic unit for each var so that the coefficients of as many terms as possible become 1
- Rewrite system of equations in terms of new dimensionless quantities
Example¶
\[ \begin{aligned} c_t = c_0 e^{-kt} &\implies (c_t/c_0) = e^{-kt} \\ 3 \text{ var} & \implies 2 \text{ var} \end{aligned} \]
Unknown Functional Form¶
Buckingham Pi Theorem