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Maths › Approximation › Regression ›

Linear

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Calculates the linear regression parameters and evaluates the regression line at arbitrary abscissas

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Contents

Interface
Class Linear
Linear Once
Page Comments

Interface

C++

Class Linear

Linear regression is a method to best fit a linear equation (straight line) of the form $\inline y(x) = a + bx$ to a collection of points $\inline (x_i,y_i) \, 1 \leq i \leq N$ , where $\inline b$ is the slope and $\inline a$ the intercept on the $\inline Y$ axis.

The algorithm basically requires minimisation of the sum of the squared distance from the data points to the proposed line. This is achieved by calculating the derivative with respect to a and b and setting these to zero.

Let us define the following

$S_x = \sum_{i = 0} ^ {N - 1} (x_i - \overline{x}) ^ 2 \qquad S_y = \sum_{i = 0} ^ {N - 1} (y_i - \overline{y}) ^ 2$

(2)

$S_{xy} = \sum_{i = 0} ^ {N - 1} (x_i - \overline{x})(y_i - \overline{y})$

(3)

Then the slope is

$b = \frac{S_{xy}}{S_{x}}$

(4)

the intercept on the Y axis

$a = \overline{y} - b \overline{x}$

(5)

Below you will find the regression graph for a set of arbitrary points, which were also used in the forthcoming example. The regression line, displayed in red, has been calculated using this class.

MISSING IMAGE!

1/reglinear-378.png cannot be found in /users/1/reglinear-378.png. Please contact the submission author.

Example 1

The following example displays the slope, Y intercept and regression coefficient for a certain set of 7 points.

#include <codecogs/maths/approximation/regression/linear.h>
#include <iostream>
#include <iomanip>
using namespace std;
 
int main() 
{
    double x[7] = { 1.5, 2.4, 3.2, 4.8,  5.0, 7.0,  8.43 };
    double y[7] = { 3.5, 5.3, 7.7, 6.2, 11.0, 9.5, 10.27 };
 
    Maths::Regression::Linear A(7, x, y);
 
    cout << "    Slope = " << A.getSlope() << endl;
    cout << "Intercept = " << A.getIntercept() << endl << endl;
 
    cout << "Regression coefficient = " << A.getCoefficient() << endl;
 
    cout << endl << "Regression line values" << endl << endl;
    for (double i = 0.0; i <= 3; i += 0.6) 
    {
        cout << "x = " << setw(3) << i << "  y = " << A.getValue(i);
        cout << endl;
    }
    return 0;
 
}

Output:

Slope = 0.904273
Intercept = 3.46212
 
Regression coefficient = 0.808257
 
Regression line values
 
x =   0  y = 3.46212
x = 0.6  y = 4.00469
x = 1.2  y = 4.54725
x = 1.8  y = 5.08981
x = 2.4  y = 5.63238
x =   3  y = 6.17494

Authors

Lucian Bentea (August 2005)

Source Code

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Members of Linear

Linear

Linear(	int	`n`
	double*	`x`
	double*	`y`	)[constructor]

Initializes the class by calculating the slope, intercept and regression coefficient based on the given constructor arguments.

Note

The slope should not be infinite.

n	The number of initial points in the arrays x and y
x	The x-coordinates of points
y	The y-coordinates of points

GetValue

doublegetValue( double x )

x	the abscissa used to evaluate the linear regression function

GetCoefficient

doublegetCoefficient(

)

The regression coefficient indicated how well linear regression fits to the original data. It is an expression of error in the fitting and is defined as:

$r = \frac{S_{xy}}{\sqrt{S_{x} \cdot S_{y}}}$

(6)

This varies from 0 (no linear trend) to 1 (perfect linear fit). If $\inline |S_y| = 0$ and $\inline |S_x| \neq 0$ , then r is considered to be equal to 1.

Linear Once

doubleLinear_once(	int	`n`
	double*	`x`
	double*	`y`
	double	`a`	)

This function implements the Linear class for one off calculations, thereby avoid the need to instantiate the Linear class yourself.

Example 2

The following graph fits a straight line to the following values:

x = 1  y = 0.22
x = 2  y = 0.04
x = 3  y = -0.13
x = 4  y = -0.17
x = 5  y = -0.04
x = 6  y = 0.09
x = 7  y = 0.11

There is an error with your graph parameters for Linear_once with options n=7 x="1 2 3 4 5 6 7" y="0.22 0.04 -0.13 -0.17 -0.04 0.09 0.11" a=1:7 .input

Error Message:Function Linear_once failed. Ensure that: Invalid C++

Parameters

n	The number of initial points in the arrays x and y
x	The x-coordinates of points
y	The y-coordinates of points
a	The x-coordinate for the output location

Returns

the interpolated y-coordinate that corresponds to a.

Source Code

Source code is available when you agree to a GP Licence or buy a Commercial Licence.

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