naginterfaces.library.correg.linregs_​const

naginterfaces.library.correg.linregs_const(x, y)

linregs_const performs a simple linear regression with dependent variable $y$ and independent variable $x$.

For full information please refer to the NAG Library document for g02ca

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g02/g02caf.html

Parameters

xfloat, array-like, shape $\left(n\right)$

$x[\text{i}-1]$ must contain $x_{\text{i}}$, for $\text{i}=1,2,\dots ,n$.

yfloat, array-like, shape $\left(n\right)$

$y[\text{i}-1]$ must contain $y_{\text{i}}$, for $\text{i}=1,2,\dots ,n$.

Returns

resultfloat, ndarray, shape $\left(20\right)$

The following information:

$result\left[0\right]$	$\overline{x}$, the mean value of the independent variable, $x$;
$result\left[1\right]$	$\overline{y}$, the mean value of the dependent variable, $y$;
$result\left[2\right]$	$s_x$ the standard deviation of the independent variable, $x$;
$result\left[3\right]$	$s_y$ the standard deviation of the dependent variable, $y$;
$result\left[4\right]$	$r$, the Pearson product-moment correlation between the independent variable $x$ and the dependent variable $y$;
$result\left[5\right]$	$b$, the regression coefficient;
$result\left[6\right]$	$a$, the regression constant;
$result\left[7\right]$	$se\left(b\right)$, the standard error of the regression coefficient;
$result\left[8\right]$	$se\left(a\right)$, the standard error of the regression constant;
$result\left[9\right]$	$t\left(b\right)$, the $t$ value for the regression coefficient;
$result\left[10\right]$	$t\left(a\right)$, the $t$ value for the regression constant;
$result\left[11\right]$	$SSR$, the sum of squares attributable to the regression;
$result\left[12\right]$	$DFR$, the degrees of freedom attributable to the regression;
$result\left[13\right]$	$MSR$, the mean square attributable to the regression;
$result\left[14\right]$	$F$, the $F$ value for the analysis of variance;
$result\left[15\right]$	$SSD$, the sum of squares of deviations about the regression;
$result\left[16\right]$	$DFD$, the degrees of freedom of deviations about the regression
$result\left[17\right]$	$MSD$, the mean square of deviations about the regression;
$result\left[18\right]$	$SST$, the total sum of squares;
$result\left[19\right]$	DFT, the total degrees of freedom.

Raises

NagValueError

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n>2$.

(errno $2$)

On entry, all $\text{n}$ values of at least one of $x$ and $y$ are identical.

Notes

In the NAG Library the traditional C interface for this routine uses a different algorithmic base. Please contact NAG if you have any questions about compatibility.

linregs_const fits a straight line of the form

$$y=a+bx$$

to the data points

$$(x_1,y_1),(x_2,y_2),\dots ,(x_n,y_n)\text{,}$$

such that

$$y_i=a+b{x}_i+e_i\text{,}i=1,2,\dots ,n(n>2)\text{.}$$

The function calculates the regression coefficient, $b$, the regression constant, $a$ (and various other statistical quantities) by minimizing

$$\sum _{i=1}^n{e}_i^2\text{.}$$

The input data consist of the $n$ pairs of observations

$$(x_1,y_1),(x_2,y_2),\dots ,(x_n,y_n)$$

on the independent variable $x$ and the dependent variable $y$.

The quantities calculated are:

1. Means:

   $$\overline{x}=\frac{1}{n}\sum _{i=1}^n{x}_i\text{;}\overline{y}=\frac{1}{n}\sum _{i=1}^n{y}_i\text{.}$$

2. Standard deviations:

   $$s_x=\sqrt{\frac{1}{n-1}\sum _{i=1}^n{(x_i-\overline{x})}^2}\text{;}{s}_y=\sqrt{\frac{1}{n-1}\sum _{i=1}^n{(y_i-\overline{y})}^2}\text{.}$$

3. Pearson product-moment correlation coefficient:

   $$r=\frac{{\sum }_{i=1}^n(x_i-\overline{x})(y_i-\overline{y})}{\sqrt{{\sum }_{i=1}^n{(x_i-\overline{x})}^2{\sum }_{i=1}^n{(y_i-\overline{y})}^2}}\text{.}$$

4. The regression coefficient, $b$, and the regression constant, $a$:

   $$b=\frac{{\sum }_{i=1}^n(x_i-\overline{x})(y_i-\overline{y})}{{\sum }_{i=1}^n{(x_i-\overline{x})}^2}\text{;}a=\overline{y}-b\overline{x}\text{.}$$

5. The sum of squares attributable to the regression, $SSR$, the sum of squares of deviations about the regression, $SSD$, and the total sum of squares, $SST$:

   $$SST=\sum _{i=1}^n{(y_i-\overline{y})}^2\text{;}SSD=\sum _{i=1}^n{(y_i-a-b{x}_i)}^2\text{;}SSR=SST-SSD\text{.}$$

6. The degrees of freedom attributable to the regression, $DFR$, the degrees of freedom of deviations about the regression, $DFD$, and the total degrees of freedom, $DFT$:

   $$DFT=n-1\text{;}DFD=n-2\text{;}DFR=1\text{.}$$

7. The mean square attributable to the regression, $MSR$, and the mean square of deviations about the regression, $MSD$:

   $$MSR=SSR/DFR\text{;}MSD=SSD/DFD\text{.}$$

8. The $F$ value for the analysis of variance:

   $$F=MSR/MSD\text{.}$$

9. The standard error of the regression coefficient, $se\left(b\right)$, and the standard error of the regression constant, $se\left(a\right)$:

   $$se\left(b\right)=\sqrt{\frac{MSD}{{\sum }_{i=1}^n{(x_i-\overline{x})}^2}}\text{;}se\left(a\right)=\sqrt{MSD(\frac{1}{n}+\frac{{\overline{x}}^2}{{\sum }_{i=1}^n{(x_i-\overline{x})}^2})}\text{.}$$

10. The $t$ value for the regression coefficient, $t\left(b\right)$, and the $t$ value for the regression constant, $t\left(a\right)$:

    $$t\left(b\right)=\frac{b}{se\left(b\right)}\text{;}t\left(a\right)=\frac{a}{se\left(a\right)}\text{.}$$

References

Draper, N R and Smith, H, 1985, Applied Regression Analysis, (2nd Edition), Wiley