Curve Fitting

Applications of numerical techniques in science and engineering often involve curve fitting of experimental data .For example ,in 1601 the German astronomer Johannes Kepler formulated the third law of planetary motion,,where is the distance to the sun measured in millions of kilometers ,T is the orbital period measured in days ,and C is a constant .The observed data pairsfor the first four planets ,Mercury ,Venus ,Earth ,and Mars ,are,and ,and the coefficient C obtained form the method of least squares is .

Least-squares Line

In science and engineering it is often the case that an experiment produces a set of data points,where the abscissas are distinct. One goal of numerical methods is to determine a formulasthat relates these variables. Usually ,a class of allowable formulas is chosen and then coefficients must be determined. There are many different possibilities for the type of function that can be used. Often there is an underlying mathematical model ,based on the physical situation , that will determine the form of the function. In this section we emphasize the class of linear functions of the form

(1) y=f(x)=Ax B

If all the numerical valuesare known to several significant digits of accuracy,then polynomial interpolation can be used successfully ; otherwise it can not . Some experiments are devised using specialized equipment so that the data points will have at least five digits of accuracy. However , many experiments are done with equipment that is reliable only to there or fewer digits of accuracy . Often there is an experimental error in the measurements ,and although three digits are recorded for the valuesand,it is realized that the true valuesatisfies

(2) ,

Whereis the measurements error .

How do we find the best linear approximation of the (1) that goes near (not always through) the points? To answer this question ,we need to discuss the errors (also called deviation or residuals):

(3) for .

There are several norms that can be used with the residuals in (3) to measure how far the curve y=f(x) lies form the data .

(4) Maximum error ,

(5) Average error ,

(6) Root-mean-square .

Example Compare the maximum error ,average error, and rms error for the linear approximationto the points and,

The error are found using the values forand

(7)

(8)

(9)

We can see that the maximum error is largest, that if one points is badly in error, its value determines . The average errorsimply averages the absolute value of the error at the various points. It is often used because it is easy to compute .The erroris often used when the statistical nature of the errors is considered.

A best-fitting line is found by minimizing one of the quantities in equations (4) through (6).Hence there are three best-fitting that we could find. The third norm is the traditional choice because it is much easier to minimize computationally.

Finding the Least-squares Line

Let be a set of N points, where the abscissasare distinct. The Least-squares Lineis the line that minimizes the root–mean- squares error .

The quantitywill be a minimum if and only if the quantity

is a minimum .The latter is visualized geometrically by minimizing the sun of the squares of the vertical distances from the points to the.

Theorem (Least-squares Line)

Suppose that are N points ,where the abscissas are distinct .The coefficients of the Least-squares Line

y=Ax B

are the solution to the following linear system , known as the normal equations :

(10)

Poof . Geometrically ,we start with the line.The vertical distance from the pointto the pointon the line is . We must minimum the sum of the squares of the vertical distance;

(11)

The minimum value of E(A,B) is determined by setting the partial derivatives andequal to zero and solving these equations for A and B .Notice that and are constants in equation (11) and that A and B are the variables ! Hold B fixed ,differentiate E(A,B) with respect to A, and get

(12)

Now hold A fixed and differentiate with respect to B and get

(13)

Curve Fitting

Data Linearization Method for

Suppose that we are given the pointsand want to fit an exponential curve of the form

(1)

The first step is to take the logarithm of both bides:

(2)

Then introduce the change of variables:

(3) , and

This results in a linear relation between the new variables X and Y:

(4)

The original pointsin the xy-plane are transformed into the points in the XY-plane .This process is called data linearization .Then the least-squares line (4) is fit to the points. The normal equations for finding A and B are

(5)

After A and B have been found ,the parameter C in equation (1) is computed:

(6)

Nonlinear least squares：

In a least squares fit, we will encounter two problems：

(1) Fitting all the functionsof the parameters to be determined by linear form,such as polynomial fitting function. This is the linear least-squares fitting problems, has solved more easily.

(2) Fitting function to be determined by all parameters can not be linear forms, such as index fitting function , which is non-linear least-squares fitting problems. Specific method described below:

For a given function of a group of measuring a few wish with nonlinear parameters of the function。

Easy to see, nonlinear least-squares fitting problem is a nonlinear function of a very small problem can be solved nonlinear optimization methods. However, when solving specific situation much

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