User:


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
Description

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 $Y$
$N$
N
Número de Mediciones
$a$
a
Operation of the Sum
$b$
b
Opposite cathetus
m
$S_{x2}$
S_x2
Suma de Productos $X^2$
$S_{xy}$
S_xy
Suma de Productos XY

Calculations

First, select the equation:   to ,  then, select the variable:   to 
Symbol
Equation
Solved
Translated

Calculations

Symbol
Equation
Solved
Translated

 Variable   Given   Calculate   Target :   Equation   To be used


Equations

Examples

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

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

be a minimum.

(ID 6890)

If it is derived

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



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

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

(ID 6891)

If it is derived

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



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

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

(ID 6892)

The regression is calculated based on which

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



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:

$ \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)}$

(ID 9441)

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

(ID 8081)

ID:(614, 0)