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


(1)代入(2)中,如式 3


image009                      Equation 3


Find the partial derivatives for a0 and  a1  respectively through functionimage020so as to make the derivatives equal to zero:


                      Equation 4


                      Equation 5


                       Equation 6


                      Equation 7



                             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;   xi  and yi are the values of any set of experiments x and y, respectively.