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ISSN: 2168-9679
Journal of Applied & Computational Mathematics
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New Square Method

Yan Ping Wang*

AiHua Computer Studio, Beijing University of Chemical, Fangshan District, Beijing, China

*Corresponding Author:
Wang YP
AiHua Computer Studi
Beijing University of Chemical
Fangshan District, Beijing, China
Tel: 110402430060
E-mail: ww_yypp@163.com

Received date: May 26, 2016; Accepted date: February 21, 2017; Published date: February 28, 2017

Citation: Wang YP (2017) New Square Method. J Appl Computat Math 6:342. doi: 10.4172/2168-9679.1000342

Copyright: © 2017 Wang YP. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

The “new square method” is an improved approach based on the “least square method”. It calculates not only the constants and coefficients but also the variables’ power values in a model in the course of data regression calculations, thus bringing about a simpler and more accurate calculation for non-linear data regression processes.

Keywords

Multi-dimensional; Non-linear; Data regression; Model application

Preface

In non-linear data regression calculations, the “least square method” is applied for mathematical substitutions and transformations in a model, but the regression results may not always be correct, for which we have made improvement on the method adopted and named the improved one as “new square method”.

Principle of New Square Method

While investigating the correlation between variables (x,y), we get a series of paired data (x1,y1,x2,y2……xn,yn) through actual measurements. Plot these data on the x–y coordinates, then a scatter diagram as shown in Figure 1 will be obtained. It can be observed that the points are in the vicinity of a curve, whose fitted equation is set as the following Equation 1 [1,2].

applied-computational-mathematics-scatter-diagram

Figure 1: Scatter diagram.

equation   (1)

where a0, a1 and k indicate any real numbers.

To establish the fitted equation, the values of a0, a1 and k need to be determined via subtracting the calculated value y from the measured value yi, i.e., via (yi–y).

Then calculate the quadratic sum of m (yi–y) as shown in Equation 2.

equation   (2)

Substitute Expression 1 into Expression 2, as shown in Expression 3:

equation   (3)

Find the partial derivatives for a0, a1 and k respectively through function Φ so as to make the derivatives equal to zero:

equation   (4)

equation   (5)

equation   (6)

Through derivation it is found that there is no analytic solution to this equation set, then computer programs are utilized to calculate its arithmetic solutions and obtain the solutions for a0, a1 and k as well as the correlation coefficient R. It is observed that the closer the correlation coefficient R is to 1, the better the model fits.

Comparison between the “New Square Method” and the “Least Square Method”

If Equation 7 as shown below is adopted to fit any data (Table 1)

equation   (7)

  Least Square Method New Square Method
Fitted Equations: y=a0+a1x equation
Calculated Regression Results: a0 and a1 a0, a1 and k

Table 1: The comparison table between the new square method and the least square method.

• In the “new square method”, the power value k of the dependent variable is calculated, while in the “least square method”, k is assumed to be 1. With the calculated power value for the dependent variable, the “new square method” is able to have the fitted equation generate a fitted line at any curve to better fit the non-linear data [3].

• In the “new square method”, non-linear data with one factor (x) can be regressed by applying the following Equation 8 in the computer programs to obtain more accurate fittings of non-linear data by regression models [4].

equation   (8)

In Equation 8:

x: Variable;

y: Function;

x,y: Dimensional (two-dimensional);

xk1,xk2,xkn: Element;

a0: Constant;

a1,a2,an: Coefficient;

k1,k2, kn: Power.

• As for the regression of non-linear data with multi-factors in the “new square method”, the following Equation 9 can be utilized in computer programs for this purpose. This equation takes into account both the contribution of factors (x1,x2……xn) to the objective function (y) and the interplays among factors (x1,x2……xn) during the regression calculation, that is why the fitted models are of high correlation.

equation   (9)

In Equation 9:

x1,x2: Variable;

y: Function;

x1,x2,y: Dimensional (three-dimensional);

equation : Element;

a0: Constant;

equation Coefficient;

equation Power.

Note: Equation 9, which takes three-dimensional data as its example, can be applied for the regression of data in curved surface data.

References

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