New Square Method

 

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.

 

I.          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”.   

 

II.       Principle of New Square Method

 

While investigating the correlation between variables, we get a series of paired data 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.  

 

Figure 1

                          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 valuefrom the measured value, i.e., via.

 

Then calculate the quadratic sum of mas 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 functionso 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. It is observed that the closer the correlation coefficient  is to 1, the better the model fits.

 

Model choose

Mechanism research Method. The method is to study the inner relation during the course. After supposing the course, set the mathematical variant among the relation of data for more than two dimensions. To making the mathematical distortion disposal to the mathematical variant, find the relating variant and objective function, and use the coefficient of the data regression computing mechanism model.

Mechanism of the method is suitable: few of data , low of Data accuracy, need mechanism model reparation the deficiencies.

Data Research Method

The method is to the two dimensions data, and to make the two dimensions data as the objective function and variant. The change Variant x makes the change of y, the change can be divided into six situations (chart 1- 6).

 Firstly, linear increasing, with the increasing of  x, the even speed of  y  increasing.

Secondly, linear reduce, with the increasing of x, the even speed of y reduce.

Thirdly, non-linear increasing, with the increasing of x, the acceleration of y increasing.

Fourthly, non-linear increasing, with the increasing of x, the deceleration of y increasing.

Fifthly, non-linear reduce, with the increasing of x, the acceleration of y reduce.

Sixthly, non-linear reduce, with the increasing of x, the deceleration of y reduce.

 

Suppose the Model of the six situations are::

                

In the first situation, when a0 > 0 , a1 > 0, k = 1

In the second situation, when a0 > 0, a1 < 0, k = 1

In the third situation, when a0 > 0 , a1 > 0, k > 1, k < 0

In the fourth situation, when a0 > 0, a1 > 0 , 0 < k < 1

In the fifth situation when a0 > 0 , a1 < 0 , 0 < k < 1

In the sixth situation when a0 > 0, a1 < 0 , k > 1 , k < 0

 

Through the above summary, if choose xk, we can select the value of k according to the relation among a0,a1,a2 and k mentioned from Situation 1 to 6. In Situation 3 and 6, the curve concave above, it is similar with exponent curve, we can select exponent form ex. In Situation 4 and 5, the curve protrudes above, it is similar with logarithm, we can select logarithm form Logx) (logarithm fundus is e)

The problem to be noticed since selecting parameters.

1.  When one variant datum is 0, the datum can't be used as divisor and get the logarithm, and we can add a number on the dimension, and make it bigger than zero.

2. When there is a negative in the datum, the datum can't be used as regression computing, multiple the dimension datum with a negative to make it bigger than zero.

3.  When get the power of some variant, it can't be too big or small, or the regression computing will intermit, sometimes the model will enlarge the error of computing