Least Squares Regression
Creates a regression polynomial of the chosen order such that the sum of the squares of the differences (residuals) between output response values predicted by the regression model and the corresponding simulation model is minimized.
Least Squares Regression Model
The least squares regression model in HyperStudy is the polynomial expression that relates the output response of interest to the factors that were varied.
Selection of the proper model is required to create an accurate approximation. However this requires a prior knowledge of the behavior of the output responses (linear, non linear, noisy, and so on) and enough runs to feed the selected model.
 Linear Regression Model

$$F(x)={a}_{0}+{a}_{1}{x}_{1}+{a}_{2}{x}_{2}+(error)$$
 Interaction Regression Model

$$F(x)={a}_{0}+{a}_{1}{x}_{1}+{a}_{2}{x}_{2}+{a}_{3}{x}_{1}{x}_{2}+(error)$$
 Quadratic Regression Model (2nd Order)

$$F(x)={a}_{0}+{a}_{1}{x}_{1}+{a}_{2}{x}_{2}+{a}_{3}{x}_{1}{x}_{2}+{a}_{4}{x}_{1}{}^{2}+{a}_{5}{x}_{2}{}^{2}+(error)$$
An approximation is only as good as the uniformity of the design sampling and, for example, a twolevel parameter only has a linear relationship in the regression. Higher order polynomials can be introduced by using more levels for the factors, but then, using more levels results in more runs.
 A linear regression model requires $n+1$ runs.
 An interaction regression model requires $\frac{(n+1)(n+2)}{2}n\text{}runs$ .
 A quadratic regression model requires $\frac{(n+1)(n+2)}{2\text{}runs}$ .
Usability Characteristics
 HyperStudy will create the least squares regression of any
order, however, in most cases polynomials of the 4th order or higher do not
increase accuracy.Note: A custom order can be defined from the Regression Terms tab.
 Suppress regression terms that are known to be insignificant.
 Residuals and diagnostics should be used to gain an understanding of the quality of the Fit.
 Quality of a Least Squares Regression Fit is a function of the number of runs, order of the polynomial, and the behavior of the application.
 If the residuals and diagnostics are not good for a Least Squares Regression
Fit, than you can increase the order of the
Fit provided you have enough runs to fit
that specific order.Note: If $n$ is the number of input variables:
 A linear model requires $n+1$ runs.
 An interaction model requires $\frac{\left(n+1\right)\left(n+2\right)}{2}n$ runs.
 A quadratic model requires $\frac{\left(n+1\right)\left(n+2\right)}{2}$ runs.
 If increasing the order does not improve the Fit quality, then you may want to inspect the input matrix collinearity and optionally add more runs. You should try the other available Fit methods as your application may have more nonlinearity than polynomials can handle.
Settings
Parameter  Default  Range  Description 

Regression Model  Linear 

