Least  Square Method 

    Least Cubic Method

    Compare of  Least Cubic Method and Least Square Method 

    Model choose

 Least Square Method 

We can usually get a series of data pairs (x₁, y₁,  x_2, y₂.....x_m, y_m), while studying relation of the two variableness. Depict the data in the right angle coordinates (as chart  1), if we find the points are near a line, we can make the straight line variant as (Formula 1-1).

Y = a₀ + a₁ X     (Formula 1-1)       

There into, a₀, a₁ and k are arbitrary real numbers 

To set the straight line variant, we should ensure a₀ and a₁, make the minimal value of the least square sum of the difference of the real observation value of Y_i and the computing value of Y using (Formula 1-1) as the optimization superior criterion.    

Order: 

   Φ =   ∑( Y_i - Y ) ²             (Formula 1-2) 

Take (Formula1-1) to (Formular1-2), then we get: :         

  Φ =   ∑( Y_i - a₀ -  a₁ X  ) ²                                   (Formula 1-3)        

When the square of  ∑( Y_i - a₀ -  a₁ X ) ²   is the smallest, we can use function Φ to get the partial differential of a₀ and a₁, and make the two partial differential to zero.         

                         (Formula 1-4)              

                      (Formula 1-5)        

Namely:              

 m a₀ + ( ∑X _i ) a₁ = Y_i                                               (Formula 1-6)       

( ∑X _i ) a₀  +   ( ∑X _i ²) a₁  =   ( ∑X _i  Y_I)                      (Formula 1-7)      

That is,to get two variant groups about the unknown numbers a₀ and a₁, solve the two groups to .

                       　　　　　　(Formula 1-8)     

 　　　　　　　      (Formula 1-9)        

get: then get  a₀ and a₁ into (Formula1-1), this time the (Formula1-1) is the regression variant linear equation, which is mathematicalematical model. 

During the period of regression, the relating formula of regression can't all be through every regression data point (x₁, y₁,  x_2, y₂.....x_m, y_m), to  estimate whether the relating formula is right, we can use correlation coefficient R, statistical variable F, and residue standard deviation S to estimate: it is better that R tends to 1, the absolute value of F is bigger and S tends to 0 . 

    (Formula 1-10)   

In the (Formula 1-10) m is sample quantity, that is the experiment times, x_i and y_i are the numerical value of experiment x and y

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   Least Cubic Method

When studying the relating relations between the two variable numbers (x, y), we can get a series of binate data(x₁, y₁,  x_2, y₂.....x_m, y_m), describing the data into the x - y orthogonal coordinate system(chart  2), the points are found near a curve. Suppose the one-variant non-linear variant of the curve such as (Formula 2-1),

Y = a₀ + a₁ X  ^k                                                   (Formula 2-1)

There into, a₀, a₁ and k are arbitrary real numbers 

To set the curve variant, the numbers of a₀, a₁ and k must be set. Use the same way with "The Least Square Method" Data Regression  based on the square sum of the deviation of the true measure value Y_i and computing value.

order:  '

Φ =   ∑( Y_i - Y ) ²                                       (Formula 2-2)

Take (Formula2-1) to  (Formula2-2) to get : 

  Φ =   ∑( Y_i - a₀ -  a₁ X ^k ) ²                             (Formula 2-3)

When the square of ∑( Y_i - a₀ -  a₁ X ^k) ²   is the smallest, we can use function Φ  to get the partial differential coefficients of a₀, a₁ and k, make the three partial differential coefficients zero.

                            (Formula 2-4)

                        (Formula 2-5)

              (Formula 2-6)

Get three variant groups about a₀,a₁ and k which are the unknown numbers, solve the groups can get mathematical model.

Also, we can judge the right of the mathematical model with the help of correlation coefficient R, statistical variable F, residual standard deviation S to judge, it is better that R tends to 1,the absolute value of F is bigger and S tends to 0. The validation is good, but error of model computing sometimes are big, to improve the mathematical model farther, the biggest error, equal error and the equal relating error of computing model are computed to validate the model.  

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  Least Cubic Method and Least  Square Method compare

Table of Least Cubic Method and Least  Square Method compare

Through the comparison, the optimize the criterion of The Least Square Method Data Regression and Least Cubic Method  ∑( Y_i - Y ) ²    are same,  Least Cubic Method accounts the power K, The Least Square Method Data Regression doesn't compute the value of k, and take it as 1.

1.Least Cubic Method use the value of computing power to make the model function curves in different rates, to draw up the curves with different curve rates. It saves the complex ways of setting mechanism model and disposing linearization and makes the regression model and data drawing up better.

2. To the regression of nonlinear data, don't use the variants of The Least Square Method Data Regression to set models to objective functions, at the same time the once regression mathematical model, since the regression , not only consider the contribution of objective function the variants take, and also consider the effects among the variants, so as to make the model correct.

3. In the Least Square Method Data Regression there is only X, In the Least Cubic Method  Theory, The variant data can have many variants X^k1,    X^{k2 .......} X^kn, i.e.(Formula 3-2). With the utility of the character, it can make the data more correct 

Y  = a₀ + a₁ X^k1 + a₂ X^k2 +...+ a_n X ^kn                   (Formula 3 - 1)

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 Model choose

1. Mechanism Study 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.

2. 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  3-1)-(chart 3-6).

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 variants of the six situations are: 

Y = a₀ + a₁ X^k          (  (Formula 4-1)

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 X^k, 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 e^X. In Situation 4 and 5, the curve protrudes above, it is similar with logarithm, we can select logarithm form LOG(X) (logarithm fundus is e)

3. 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

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