naginterfaces.library.correg.linregm_​coeffs_​noconst

naginterfaces.library.correg.linregm_coeffs_noconst(n, sspz, rz)

linregm_coeffs_noconst performs a multiple linear regression with no constant on a set of variables whose sums of squares and cross-products about zero and correlation-like coefficients are given.

For full information please refer to the NAG Library document for g02ch

https://support.nag.com/numeric/nl/nagdoc_30.1/flhtml/g02/g02chf.html

Parameters

nint

$n$, the number of cases used in calculating the sums of squares and cross-products and correlation-like coefficients.

sspzfloat, array-like, shape $(k+1,k+1)$

$sspz[\text{i}-1,\text{j}-1]$ must be set to ${\tilde{S}}_{\text{i}\text{j}}$, the sum of cross-products about zero for the $\text{i}$th and $\text{j}$th variables, for $\text{j}=1,2,\dots ,k+1$, for $\text{i}=1,2,\dots ,k+1$; terms involving the dependent variable appear in row $k+1$ and column $k+1$.

rzfloat, array-like, shape $(k+1,k+1)$

$rz[\text{i}-1,\text{j}-1]$ must be set to ${\tilde{R}}_{\text{i}\text{j}}$, the correlation-like coefficient for the $\text{i}$th and $\text{j}$th variables, for $\text{j}=1,2,\dots ,k+1$, for $\text{i}=1,2,\dots ,k+1$; coefficients involving the dependent variable appear in row $k+1$ and column $k+1$.

Returns

resultfloat, ndarray, shape $\left(13\right)$

The following information:

$result\left[0\right]$	$SSR$, the sum of squares attributable to the regression;
$result\left[1\right]$	$DFR$, the degrees of freedom attributable to the regression;
$result\left[2\right]$	$MSR$, the mean square attributable to the regression;
$result\left[3\right]$	$F$, the $F$ value for the analysis of variance;
$result\left[4\right]$	$SSD$, the sum of squares of deviations about the regression;
$result\left[5\right]$	$DFD$, the degrees of freedom of deviations about the regression;
$result\left[6\right]$	$MSD$, the mean square of deviations about the regression;
$result\left[7\right]$	$SST$, the total sum of squares;
$result\left[8\right]$	$DFT$, the total degrees of freedom;
$result\left[9\right]$	$s$, the standard error estimate;
$result\left[10\right]$	$R$, the coefficient of multiple correlation;
$result\left[11\right]$	$R^2$, the coefficient of multiple determination;
$result\left[12\right]$	${\overline{R}}^2$, the coefficient of multiple determination corrected for the degrees of freedom.

coeffloat, ndarray, shape $(k,3)$

For $i=1,2,\dots ,k$, the following information:

$coef[i-1,0]$

$b_i$, the regression coefficient for the $i$th variable.

$coef[i-1,1]$

$se\left(b_i\right)$, the standard error of the regression coefficient for the $i$th variable.

$coef[i-1,2]$

$t\left(b_i\right)$, the $t$ value of the regression coefficient for the $i$th variable.

rznvfloat, ndarray, shape $(k,k)$

The inverse of the matrix of correlation-like coefficients for the independent variables; that is, the inverse of the matrix consisting of the first $k$ rows and columns of $rz$.

czfloat, ndarray, shape $(k,k)$

The modified inverse matrix, $C$, where

$$cz[i-1,j-1]=\frac{rz[i-1,j-1]\times rznv[i-1,j-1]}{sspz[i-1,j-1]}\text{,}i,j=1,2,\dots ,k\text{.}$$

Raises

NagValueError

(errno $1$)

On entry, $k=⟨value⟩$.

Constraint: $k\ge 1$.

(errno $3$)

On entry, $n=⟨value⟩$ and $k=⟨value⟩$.

Constraint: $n\ge k+1$.

(errno $5$)

The $\text{K}\times \text{K}$ partition of $rz$ which requires inversion is not positive definite.

(errno $6$)

The refinement following the actual inversion has failed.

Notes

No equivalent traditional C interface for this routine exists in the NAG Library.

linregm_coeffs_noconst fits a curve of the form

$$y=b_1{x}_1+b_2{x}_2+\cdots +b_k{x}_k$$

to the data points

$$\begin{array}{c}\begin{array}{c}(x_{11},x_{21},\dots ,x_{k1},y_1)\\ (x_{12},x_{22},\dots ,x_{k2},y_2)\\ ⋮\\ (x_{1n},x_{2n},\dots ,x_{kn},y_n)\end{array}\end{array}$$

such that

$$y_i=b_1{x}_{1i}+b_2{x}_{2i}+\cdots +b_k{x}_{ki}+e_i\text{,}i=1,2,\dots ,n\text{.}$$

The function calculates the regression coefficients, $b_1,b_2,\dots ,b_k$, (and various other statistical quantities) by minimizing

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

The actual data values $(x_{1i},x_{2i},\dots ,x_{ki},y_i)$ are not provided as input to the function. Instead, input to the function consists of:

1. The number of cases, $n$, on which the regression is based.

2. The total number of variables, dependent and independent, in the regression, $(k+1)$.

3. The number of independent variables in the regression, $k$.

4. The $(k+1)\times (k+1)$ matrix $\left[{\tilde{S}}_{ij}\right]$ of sums of squares and cross-products about zero of all the variables in the regression; the terms involving the dependent variable, $y$, appear in the $(k+1)$th row and column.

5. The $(k+1)\times (k+1)$ matrix $\left[{\tilde{R}}_{ij}\right]$ of correlation-like coefficients for all the variables in the regression; the correlations involving the dependent variable, $y$, appear in the $(k+1)$th row and column.

The quantities calculated are:

1. The inverse of the $k\times k$ partition of the matrix of correlation-like coefficients, $\left[{\tilde{R}}_{ij}\right]$, involving only the independent variables. The inverse is obtained using an accurate method which assumes that this sub-matrix is positive definite (see Further Comments).

2. The modified matrix, $C=\left[c_{ij}\right]$, where

   $$c_{ij}=\frac{{\tilde{R}}_{ij}{\tilde{r}}^{ij}}{{\tilde{S}}_{ij}}\text{,}i,j=1,2,\dots ,k\text{,}$$

   where ${\tilde{r}}^{ij}$ is the $(i,j)$th element of the inverse matrix of $\left[{\tilde{R}}_{ij}\right]$ as described in (a) above. Each element of $C$ is thus the corresponding element of the matrix of correlation-like coefficients multiplied by the corresponding element of the inverse of this matrix, divided by the corresponding element of the matrix of sums of squares and cross-products about zero.

3. The regression coefficients:

   $$b_i=\sum _{j=1}^k{c}_{ij}{\tilde{S}}_{j(k+1)}\text{,}i=1,2,\dots ,k\text{,}$$

   where ${\tilde{S}}_{j(k+1)}$ is the sum of cross-products about zero for the independent variable $x_j$ and the dependent variable $y$.

4. 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, $SST$:

   $SST={\tilde{S}}_{(k+1)(k+1)}$, the sum of squares about zero for the dependent variable, $y$;

   $SSR={\sum }_{j=1}^k{b}_j{\tilde{S}}_{j(k+1)}\text{;}SSD=SST-SSR$.

5. 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, $DFT$:

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

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

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

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

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

8. The standard error estimate:

   $$s=\sqrt{MSD}\text{.}$$

9. The coefficient of multiple correlation, $R$, the coefficient of multiple determination, $R^2$, and the coefficient of multiple determination corrected for the degrees of freedom, ${\overline{R}}^2$:

   $$R=\sqrt{1-\frac{SSD}{SST}}\text{;}{R}^2=1-\frac{SSD}{SST}\text{;}{\overline{R}}^2=1-\frac{SSD\times DFT}{SST\times DFD}\text{.}$$

10. The standard error of the regression coefficients:

    $$se\left(b_i\right)=\sqrt{MSD\times c_{ii}}\text{,}i=1,2,\dots ,k\text{.}$$

11. The $t$ values for the regression coefficients:

    $$t\left(b_i\right)=\frac{b_i}{se\left(b_i\right)}\text{,}i=1,2,\dots ,k\text{.}$$

References

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