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.
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 (x_{1},y_{1},x_{2},y_{2}……x_{n},y_{n}) 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.
where a_{0}, a_{1} and k indicate any real numbers.
To establish the fitted equation, the values of a_{0}, a_{1} and k need to be determined via subtracting the calculated value y from the measured value y_{i}, i.e., via (y_{i}–y).
Then calculate the quadratic sum of m (y_{i}–y) as shown in Equation 2.
Substitute Expression 1 into Expression 2, as shown in Expression 3:
Find the partial derivatives for a_{0}, a_{1} and k respectively through function Φ so as to make the derivatives equal to zero:
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 a_{0}, a_{1} 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)
Table 1: The comparison table between the new square method and the least square method.
In Equation 8:
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 (x_{1},x_{2}……x_{n}) to the objective function (y) and the interplays among factors (x_{1},x_{2}……x_{n}) during the regression calculation, that is why the fitted models are of high correlation.
In Equation 9:
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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Comment
Mathematicians would call this Nonlinear Least Squares (the model is non-linear in some of the parameters - here the exponents), vs Linear Least Squares (which is linear in ALL the parameters). A commonly recommended algorithm for solving such problems is the Levenberg-Marquardt Algorithm (see e.g. [1,2,3]).
[1] "Levenberg–Marquardt algorithm". Wikipedia (https://en.wikipedia.org/wiki/Levenberg%E2%80%93Marquardt_algorithm)
[2] "Numerical Optimization" Nocedal, Jorge, and Stephen Wright. Springer Science & Business Media, 2006.
[3] "Numerical Optimization. Lecture Notes 23 - 'Nonlinear Least Squares Problems - Algorithms'". P Blomgren (2018), http://terminus.sdsu.edu/SDSU/Math693a/Lectures/23/lecture.pdf
He does not need identity transformation and substitution. There is no analytic solution for the power value, so it can only be solved numerically by computer.
How does it compare to linear regression where the variables are transformed using a power transformation, or to polynomial regression?
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