Curve Fitting

Part 6: Summary

1. Suppose you have n pairs of data points,
   (T₁,Y₁), (T₂,Y₂), ..., (T_n,Y_n),
   and suppose you want to fit a cubic polynomial

   y = a + b t + c t² + d t³

   
   to this data. In what vector space can this be formulated as a linear algebra problem? What linear algebra problem is equivalent to finding the coefficients a, b, c, and d of the least squares cubic?
   
   

2. For the cubic fit in question 1, how many normal equations are there and how many variables do they have? What are the dimensions of the matrix X associated with the normal equations? How is matrix X constructed?
   
   

3. Suppose that, instead of a cubic polynomial, we want to fit an exponential function

   y = a e^b t

   
   to the data of question 1. What function S of the parameters a and b needs to be minimized to find the best least squares fit? Why can't we just find the optimal values of a and b by solving the normal equations associated with fitting

   ln(y) = ln(a) + b t ?

   
   
4. In your own words, describe the steps of the iterative process needed to solve the nonlinear least squares problem of problem 3.
   

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