INSITTUTO DE ESTUDIOS SUPERIORES DE TAMAPULIPAS
RED DE UNIVERSIDADES AN?HUAC 

M?XICO MMVI 

Polynomial Regression through Least Square Method 

Prof. David Macias Ferrer
david.macias@iest.edu.mx
Madero City, Mexico
http://www.geocities.com/dmacias_iest/MyPage.html 

Goals 

 

• To approximate a Points Dispersion through Least Square Method using a Quadratic Regression Polynomials and the Maple Regression Commands.

 

• To show the powerful Maple 10 graphics tools to visualize the convergence of this Polynomials.

 

Least Square Method using a Regression Polynomials 

 

Let [] ∀k∈ℕ be a dispersion point in . A general regression polynomials is given by: 

 

where etc. According the Least Square principle, the coefficient can be determined by: 

 

Application 

 

Problem: Supose that we have the follow points dispersion: 

 

To use the Maple Tools to find a Quadratic Regression Polynomials to aproximate the dispersion using Least Square Method. 

According the Least Square Method, the Regresion Polynomials of second degree is given by: 

 

 

where . Then, the coefficient are given by: 

 

 

 

 

 

The following spreadsheet showing the points dispersion: 

 

 

 

In this spreadsheet, the group of yellow cells represent the points dispersion; H2 represent ; H3 and I2 represent ;  J2, I3 and H4 represent  ; J3 and I4 represent ; J4 represent , (see green cells). On the other hand, K2 represent ; K3 represent ; finally, the cell K4 represent  ,(see cyan cells).  

 

Using the Spread Tools, we have: 

 

 

 

(3.1)

 

 

 

(3.2)

 

 

 

The coefficients of Quadratic Polynomials are given by: 

 

 

(3.3)

 

 

(3.4)

 

 

(3.5)

 

 

(3.6)

 

Finally, the Quadratic Polynomials is: 

 

 

(3.7)

 

 

The above mentioned, shows the habitual procedure in a typical class of numeric methods in engineering, this is, what the student has to make in a written exam. But, in an applied problem of science, the Maple Tools can simplify the solution process. 

 

Now, we have the Statistics Package to find the quadratic polynomials 

 

 

 

 

 

 

(3.8)

 

 

(3.9)

 

Let us see the residual sum of squares and the standard errors: 

 

 

(3.10)

 

 

Let us see the graphics of Points Dispersion, p and QuadPoly: 

 

 

 

 

 

Bibliography 

Burden R, Faires D., "Numerical Analysis", Fifth Edition, USA, PWS Publishing Company, 1993 

 

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