Team:XH-China/Model
Polynomial fitting model
Polynomial regression fitting model is employed to predict the function of absorbance and tyrosine. The R2 and SSE are used to evaluate the accuracy of model, and the result indicates the model established in this paper has high accuracy. Therefore, it is applied to predict the value of tyrosine according the absorbance.
Least squares polynomial curve fitting, according to the given m points, does not require the curve to pass through these points accurately, but the approximate curve y= φ(x) of the curve y=f(x). Given a data point pi(xi,yi), where i=1,2,...,m. Find the approximate curve y= φ(x) and minimize the deviation of the approximate curve from y=f(x). The deviation of the approximate curve at point pi is δi= φ(xi)-y, i=1,2,...,m.
step 1
The fitting polynomial is:
step 2
The sum of the distances from each point to this curve, which means the sum of squared deviations is:
step 3
In order to obtain the value of a that meets the conditions, calculate the partial derivative of ai on the right side of the equation:
step 4
Simplify the left side of the equation:
step 5
Expressing these equations in the form of a matrix:
step 6
That is,X × A = Y. After the linear equation is solved, the coefficient matrix of the polynomial of the fitted curve can be obtained.
The absorbance OD660 was used as the independent variable x and tyrosine was used as the dependent variable y to construct a fourth-order polynomial regression fitting model.
The data used to fit the model is listed as follows:
| Content micromolar | Absorbance value OD660 |
|---|---|
| 0.055 | 0.173 |
| 0.111 | 0.261 |
| 0.221 | 0.367 |
| 0.442 | 0.553 |
| 0.553 | 0.648 |
We used MATLAB to solve the coefficient of polynomial fitting, and respectively tested the polynomial to the highest power of 1, 2, 3, 4, 5 and 6. The results showed that the fitting accuracy obtained by the fourth power was the best, so the fourth power was selected as the highest power of polynomial fitting, and the results were illustrated below:
Linear model Poly4:
Considering Coefficients are:
p1 = 5.423
p2 = -11.43
p3 = 8.855
p4 = -1.8
p5 = 0.1558
The Figure 1 shows the fitting curve, which fits well with the real data points.
In order to test the accuracy of the fitted model, select SSE (Sum of Squares due to Error), R2 coefficient of determination, coefficient of Determination and RMSE (Root Mean Squared Error). The formulas are as follows:
Where y is the true value, y is the predicted value, y is the average value, and n y is the number of samples.
| Error test index | Value |
|---|---|
| SSE | 0 |
| R2 | 1 |
| RMSE | 0 |
It can be concluded from the above table that the precision of the fourth-order polynomial model established in this paper is extremely high.
| blank | 20ul keratinase | 50ul keratinase | 100ul keratinase | |
|---|---|---|---|---|
| OD660 | 0.003 | 0.323 | 0.456 | 0.588 |
| Tyrosine conten（uMol） | 0.1505 | 0.1721 | 0.3270 | 0.4835 |