Approximation (Exam Spring 1999, Question 5)

Usage notes

a: Find an approximation function

Several data points y[k] are observed for different times t[k]. Furthermore, previous experience shows that as |t| approaches infinity, the values converge to a limit, as is happening with the data shown below:

  t[k] 	1 	2 	 3 	 4 	 5 	 6 	7 	8 	 9
y[k] 	-2 	53 	70 	83 	95 	102 	96 	101 	98

It may be helpful to use the following graphical representation of the observed data points. Note that the linear interpolation is shown for purely aesthetic reasons.

Suggest an approximation function for y, that is a formula y=f(t) which will satisfy approximately the observations, and which is essentially different from the function given below (it should not contain an exponential function). Note that f is expected to contain some parameters which may be determined later. Example: "a*t^2 + b*t + c"

y = f(t) = ___________________

Maple will try to find the optimal parameter values by doing a least-squares fit. To minimize the sum of error squares, it will set the partial derivatives (with respect to the parameters) to zero. Note that solving this equation system will give us local minima as well as local maxima and other extremal points. Further, there may be several local minima. In order to get a useful fit you may have to assist Maple and estimate some of your parameters. Enter none, one or several estimates here. Example: "a=3.5, c=12"
___________________

Maple has found that the following parameter values describe a local extremal point: not available The sum of error squares is: not available

b: The equation system

From now on, the approximation function is assumed to be f(t)=a+b*exp(c*t).

Write down the expression which has to be minimized in order to determine the optimal values for a, b and c. You can refer to the observed data with t[k] or y[k], where 1<=k<=9. Example: "sum(a*t[k]+y[k], k=1..9)"

S(a, b, c) = ___________________

Write down the equations to solve to determine the parameters a, b and c, such that the approximation to the given data is optimal in the least squares sense. Example: "sum(a*y[k], k=0..9) = 42"

Equation 1: ___________________
Equation 2: ___________________
Equation 3: ___________________

c: Solving the equation system

If you had to eliminate as many unknowns as possible from the above equation system, which unknowns would you eliminate? Select none, one or several parameters.

a

b

c

Solving the reduced equation system, we get a set of parameter values a*, b* and c*. What do we have to do now?

Nothing. Using the parameters a=a*, b=b* and c=c* in f(t), the approximation error is locally minimized.

(a*,b*,c*) is not necessarily a local minimum. We have to show that the Hessian of S at (a*,b*,c*) has only negative eigenvalues.

Same as above, but all eigenvalues have to be positive.

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