Druyd:Knowledge

Straight

Storyboard

>Model

ID:(614, 0)

Simulator

Definition

The attached demo allows you to make a least-squares adjustment of a line.

ID:(8081, 0)

Straight

Storyboard

Variables

Symbol

Text

Variable

Value

Units

Calculate

MKS Value

MKS Units

$x_0$

x_0

Coordenada fija en $X$

$y_0$

y_0

Coordenada fija en $Y$

$N$

Número de Mediciones

$a$

Operation of the Sum

$b$

Opposite cathetus

$S_{x2}$

S_x2

Suma de Productos $X^2$

$S_{xy}$

S_xy

Suma de Productos XY

Calculations

Equation

Number

First, select the equation:

, then, select the variable:

Symbol

Equation

Solved

Translated

Calculations

Symbol

Equation

Solved

Translated

Variable

Given

Calculate

Target :

Equation

To be used

Equations

$\sum_i (y_i-ax_i-b)^2 = min$

sum_i (y_i-ax_i-b)^2 = min

$a=\displaystyle\frac{NS_{xy}-S_xS_y}{NS_{xx}-S_x^2}$

a =( N * S_xy - S_x * S_y )/( N * S_xx - S_x ^2)

$ b =\displaystyle\frac{ S_{x2} S_y - S_x S_{xy}}{ N S_{x2} - S_x ^2}$

b =( S_x2 * S_y - S_x * S_xy )/( N * S_x2 - S_x ^2)

$ \epsilon =\displaystyle\frac{(( N S_{x2} - S_x ^2) S_{y2} - S_{x2} S_y ^2+2 S_x S_{xy} S_y - N S_{xy} ^2}{ N ( N S_{x2} - S_x ^2)}$

epsilon =(( N * S_x2 - S_x ^2)* S_y2 - S_x2 * S_y ^2+2* S_x * S_xy * S_y - N * S_xy ^2)/( N * S_x2 - S_x ^2)

Examples

Straight line

To adjust data (x_i, y_i) to a line of the type

y = ax + b

you must calculate the values a and b such that the difference of the squares

equation

be a minimum.

Slope

If it is derived

equation=6890

with respect to a and the result is equal to zero the equation is obtained:

S_{xy}+aS_{x2}+bS_x=0

where

S_x=\sum_ix_i, S_{x2}=\sum_ix_i^2 and S_{xy}=\sum_ix_iy_i

If the operation is repeated for b the equation is obtained:

bN-S_y+aS_x=0

with S_y=\sum_iy_i.

The solution of the equations leads to the slope being

equation

Constant

If it is derived

equation=6890

with respect to a and the result is equal to zero the equation is obtained:

S_{xy}+aS_{x2}+bS_x=0

where

S_{x,n,y,m}=\sum_ix_i^ny_i^m

that in the case that n or m are zero the factor x or y is not written and in the case of the unit the number is not included.

If the operation is repeated for b the equation is obtained:

bN-S_y+aS_x=0

with S_y=\sum_iy_i.

The solution of the equations leads to the constant being

equation

Deviation

The regression is calculated based on which

equation=6890

be a minimum. If the square is developed and the root of this value is divided by the mean value, a measure of the mean deviation of the regression is obtained:

equation

Simulator

The attached demo allows you to make a least-squares adjustment of a line.

>Model

ID:(614, 0)