Problem 22    Due 4/3/00

In this problem we solve a combined linear and
non-linear least squares regression problem.
We are trying to look at the growth of bacterial
cells over time.  There are two species present
in our nutrient broth, each with different growth
rates and initial concentrations, however we can't
easily distinguish between them.  Instead we measure
the total concentration of cells as a function of
time, and try to infer the growth rate of each of
the two species.  Thus:

   Total = A * exp (ka*t) + B * exp (kb*t)

We have the observed data on total cell concentration
given below.  We wish to use least-squares regression
to calculate the two growth rate constants.  Note that
you cannot linearize the above equation!  Solve this
problem by formulating it in the least squares sense
(e.g., the sum of the square of the deviation between
the model prediction for A, B, ka, and kb and the
observed data points).  Since the problem is linear in
the parameters A and B, write a function which returns
the optimum values of these parameters and the sum of
squares for fixed ka and kb.  This will be a linear least
squares problem.  Then write an outer program which
does the non-linear optimization for ka and kb.  This
is much faster than solving the problem as a four parameter
non-linear optimization problem.  You may use the matlab
routine fmins for the two-parameter non-linear optimization
problem.  Type help fmins to learn more about its use.

In addition to getting the best fit values of the parameters,
determine the matrix of covariance.  This is most easily
done by 1) using the deviation between the data points and
the fitted model to calculate the variance in the data
measurements themselves, 2) determining the change in the
best fit parameter values with changes in each of the data
points (e.g., take the derivative grad f where f is the
array of fitted parameter values and the gradient is with
respect to each of the n data points.  Grad f is thus a
4 x n matrix, which is computed numerically), and 3) use
the dependence matrix and the data variance to calculate the
matrix of covariance of the parameters.  This will only work
if the variance and / or the dependence is small (the penalty
of ignoring higher order terms when calculating the variance).

What are the 95% confidence limits of the two growth rates?

Hint:  One of the growth constants will be negative (the
cells are dying off).

Hint:  When computing the dependence of the fitting parameters
on the data, don't vary the data by too small amount, since the
routine fmin has a preset target precision.  If you make too small
a change, then you can't compute the derivative you need!  I found
that a change in data scaled with the standard deviation of the
data was effective in getting grad f.

The data is:

       t         c
         0    4.1077
    0.0500    3.8447
    0.1000    3.5597
    0.1500    3.2212
    0.2000    3.0120
    0.2500    2.8910
    0.3000    2.8429
    0.3500    2.3134
    0.4000    2.3515
    0.4500    2.5100
    0.5000    2.4260
    0.5500    2.3457
    0.6000    2.4561
    0.6500    2.3981
    0.7000    2.5175
    0.7500    2.8218
    0.8000    3.1638
    0.8500    3.3671
    0.9000    3.5864
    0.9500    4.1444
    1.0000    4.5336