Fitting a Curve to a Data Set with the New Gradient Operator in Mathcad

Fitting a Curve to a Data Set with the New Gradient Operator

This article explains how to use the Mathcad function genfit to fit a general, nonlinear function to a data set. It also shows how to use genfit with the new gradient operator, introduced in Mathcad 14, to obtain a better fit in certain cases.

Fitting a Curve to a Data Set Curve fitting - or regression analysis - is the process of finding a curve that best fits a set of data points. Typically, you specify the curve by a model function with unknown parameters. Your goal is to find the values of the parameters that minimize the overall difference between the data values and the corresponding function values. Start by looking at an example that uses the data in the table below:

data  01 0 00 1 0.2 0.1546 2 0.4 0.49 3 0.6 1.0249 4 0.8 1.3897 5 1 1.9376 6 1.2 3.3416 7 1.4 4.836 8 1.6 5.9844 9 1.8 6.1197 10 2 8.4171

The first column contains the x-values of the data, while the second column contains the y-values.

  X data 0 Y data 1 Here is a plot of the data in the xy-plane. Each point corresponds to a pair of numbers (x, y) in a single row of the table.

6 Y 4

0 0 0.5 1 1.5 2 X

Suppose you believe that the data is modeled by a model function of the b form Ax , where x is a variable and A and b are unknown parameters. You can use the Mathcad function genfit to fit a nonlinear function like b Ax to the data.The function genfit has the following form: genfit() X Y guess f where  X and Y are vectors containing the x-values and the y-values, respectively, of the data.  guess is a vector of initial guess values for the parameters.  f is the model function.

In this example, define f as

b fxA()Ax b  

Notice that the parameters A and b are listed after x as arguments to the function. Next, create a vector of guess values for A and b:

5  guess    2 

The first entry is the guess for A and the second is the guess for b. Now apply genfit to return a vector containing the parameters A and b that give the best fit.

A     genfit() X Y guess f  b 

A 2.374

b 1.799

b The graph of the function fxA()Ax b =  with these parameter values is shown below, along with the original data points.

10 10

8 8

6 6 YfxA() b 4 4

2 2

0 0 0 0.5 1 1.5 2 Xx Using the Mathcad 14 Gradient Operator to Improve the Fit

In some cases, you can improve the results of genfit by supplying it with the partial derivatives of the model function with respect to the unknown parameters. This is often the case when the model function is a rational function. The partial derivatives help guide the genfit algorithm toward the optimal solution. The gradient operator provides an easy way to compute the partial derivatives.

Here's an example.

data1  01 0 -0.6 0.526 1 -0.5 0.369 2 -0.4 0.3637 3 -0.3 0.2878 4 -0.2 0.2352 5 -0.1 0.2197 6 0 0.2347 7 0.1 0.2241 8 0.2 0.2212 9 0.3 0.2199 10 0.4 0.2056

  X1 data1 0 Y1  data1 1

You can model the data with a rational function of the form

a0  a1  x g()xa  a2  a3  x in which the unknown parameters a0, a1, a2, anda3 are contained in a single vector a. Note that g is undefined when its denominator is 0. Now apply genfit as in the previous example.

1    1 guess1    2    2 

badparams genfit() X1 Y1 guess1 g

As you can see from the graph below, the resulting parameters do not give a very good fit.

0.6 0.5 0.5 0.4 0.4 Y1 g() x badparams 0.3 0.3

0.2 0.2

 0.4  0.2 0 0.2 0.4 X1 x To get a better fit, you can supply the last argument to genfit as a vector G, whose first entry is the model function g, and whose remaining entries are the partial derivatives of g with respect to the unknown parameters.

The Gradient Operator To compute the partial derivatives of g, press [Ctrl] [Shift] G to insert the Mathcad 14 gradient operator.

∇

Next, type g(x,a), the model function, in the upper-right placeholder, and type a, the vector of parameters, in the lower-left placeholder.

∇a gxa() Press [Ctrl] [Shift] [.] to evaluate the result symbolically.  1   xa 3  a2     x   xa 3  a2  ∇ gxa()    a xa 1  a0     2  xa 3  a2    xxa  1  a0    2   xa 3  a2 

The result is a vector containing the partial derivatives of g with respect to the unknown parameters in a. Note that the vector does not contain the partial derivative of g with respect to x.

To create the vector G for genfit, use the stack function to put the model function g on top of the vector of partial derivatives.

G(xa ) stack g() x a ∇a gxa()

 xa 1  a0   xa  a   3 2   1   xa 3  a2     x  Gxa()   xa 3  a2    xa 1  a0     2  xa 3  a2    xxa  1  a0    2   xa 3  a2  Now, apply genfit with the vector G as its last argument as follows: 1. Type

goodparams genfit() X1 Y1 guess1 G

2. Right-click the function genfit and select Levenberg Marquardt from the drop-down menu.

Note: For this problem, the Levenberg Marquardt algorithm is more effective than the default algorithm, Optimized Levenberg Marquardt. Unlike the default algorithm, you must supply the partial derivatives when using Levenberg Marquardt. The graph below clearly shows that the second set of parameters, goodparams, provides a better fit to the data than badparams:

0.5 0.5 0.4 0.4 g() x badparams Y1 0.3 g() x goodparams 0.3

0.2 0.2

0.1 0.1  0.4  0.2 0 0.2 0.4 X1 x You can confirm that goodparams provides a better fit by computing the sums of squares of the residuals for the two fits.

goodresiduals Y1 g() X1 goodparams badresiduals Y1 g() X1 badparams

i010 

2 2  goodresidualsi  0.004  badresidualsi  0.083 i i