least 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, and a₁ 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

把(式1)代入(式2)中，如式 3：

                      Equation 3

Find the partial derivatives for a₀ and  a₁  respectively through functionso as to make the derivatives equal to zero:

                      Equation 4

                      Equation 5

Derivation： 

                       Equation 6

                      Equation 7

Derivation：  

                             Equation  8

                      Equation 9

 Obtain the solutions for a₀ and a₁ as well as the correlation coefficient. It is observed that the closer the correlation coefficient R is to 1, the better the model fits.

            Equation 10

In (Equation 10), m is the sample size, that is, the number of experiments;   x_i  and y_i are the values of any set of experiments x and y, respectively.