An artificial data set for linear regression. — data_numeric

A data.frame with numeric response and explanatory variables simulated the real life borehole example of the flow rate of water through a borehole from an upper aquifer to a lower aquifer separated by an impermeable rock layer.

Usage

data_numeric_regression

Format

`data_numeric_regression`

A data frame with 10000 rows and 9 columns:

$\mathcal{Y}$: response variable, flow rate through the borehole.
$r_{\mathrm{w}}$: the radius of borehole.
$r$: the radius of influence.
$T_{\mathrm{u}}$: the transmissivity of upper aquifer.
$T_1$: the transmissivity of lower aquifer.
$H_{\mathrm{u}}$: the potentiometric head of upper aquifer.
$H_1$: the potentiometric head of lower aquifer.
$L$: the length of borehole.
$K_{\mathrm{w}}$: the hydraulic conductivity of borehole.

Details

The response variable $\mathcal{Y}$, the flow rate through the borehole in $m^3 / yr$, is determined by a complex nonlinear function as follows, $$\mathcal{Y}=\frac{2 \pi T_{\mathrm{u}}\left(H_{\mathrm{u}}-H_{\mathrm{l}}\right)}{\ln \left(r / r_{\mathrm{w}}\right)\left[1+\frac{2 L T_{\mathrm{u}}}{\ln \left(r / r_{\mathrm{w}}\right) r_{\mathrm{w}}^2 K_{\mathrm{w}}^2}+\frac{T_{\mathrm{u}}}{T_1}\right]},$$ where the 8 input variables with their usual input ranges are listed as follows:

$r_{\mathrm{w}} \in[0.05,0.15]$ means the radius of borehole ($m$);
$r \in[100,50000]$ means the radius of influence ($m$);
$T_{\mathrm{u}} \in[63070,115600]$ means the transmissivity of upper aquifer($\left(\mathrm{m}^2 / \mathrm{yr}\right)$);
$T_1 \in[63.1,116]$ means the transmissivity of lower aquifer($\left(m^2 / y r\right)$);
$H_{\mathrm{u}} \in[990,1110]$ means the potentiometric head of upper aquifer($m$);
$H_1 \in [700,820]$ means the potentiometric head of lower aquifer ($m$);
$L \in[1120,1680]$ means the length of borehole ($m$);
$K_{\mathrm{w}} \in[9855,12045]$ means the hydraulic conductivity of borehole ($m / y r$).

The distribution of $r_{\mathrm{w}}$ is the normal distribution$\mathcal{N}\left(0.10,0.0161812^2\right)$, the distribution of $r$ is the lognormal distribution $\operatorname{Lognormal}\left(7.71,1.0056^2\right)$, and the distributions of other variables are all continuous uniform distribution on their corresponding domains.