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									                                     CURVE FITTING

There are three main components to science and engineering: Theory, computer simulations and
experiments. In the case of experiments, it is necessary to be able to interpret the data and to
extract useful information. Curve fitting is a very powerful tool in that it allows finding the
relationships between variables.

For example, properly fitting the data of experiments measuring the drag force on an object as a
function of wind velocity can produce the very useful result that the drag is roughly a function of
the velocity squared (see figure 1):

                                F  .2741 V 1.9842

                             Figure 1: Curve fitting for drag .vs. wind velocity

The goal of curve fitting is therefore to develop some tools that can help approximate data points
with useful functions. The first useful function one can think of is a linear curve, i.e. how to fit a
straight line to data points. Since the data may not be naturally aligned, one needs to design a
“best fit”. For example, figure 2 depicts a linear fit to the data relating the drag force to the wind
velocity. This is the “best fit” in the sense that the sum of the square of the error between the fit
and the data is minimized. This is called a least square fit.
                          Figure 2: Linear fit to the data drag .vs. wind velocity

A similar mathematical construct exists for defining “quadratic best fit” or “cubic best fit”, etc.
The tools function in MATLAB is quite useful in this regard. ToolsBasic Fitting was used to
produce the results in figure 3.

                         In class: Description of ToolsBasic Fitting

                                Description of polyfit and polyval.
                            Figure 3: Different fitting curves for the same data

However, blind curve fitting like those in figure 3 is not always useful. For example, in the case
of the drag-versus-velocity example, we need to impose that the drag force be zero when the
wind velocity is zero (otherwise, the fit will have little physical meaning). In this case, trying a
linear or a quadratic or a cubic fit will not produce acceptable results since the intercepts will not
be equal to zero in any of those cases. A better approach is to use a transform that will convert
the data into a more useable form. This is called linearization in this case and there are three
standard linearization strategies (note that there exits other models, but the following three are
very informative):

   1. The exponential model:

                                                  y  aebx
   2. The power model:

                                                   y  ax b
   3. The saturation-growth-rate model:

                                                      b x
Notes: It is useful to keep in mind the following when deciding which model to use: In the case
of the exponential model, y is different from zero when x=0. Therefore this model is not well-
suited for a case like the drag .vs. wind velocity relation. For the other two models we have y=0
when x=0 but the difference between the two is that the saturation-growth-rate model “levels
off” as x increases, modeling a limiting condition of growth (or decay).

The linearization is obtained in the following way. Instead of expressing y as a function of x we

   1. ln(y) as a function of x in the case of the exponential model.

   2. ln(y) as a function of ln(x) in the case of the power model.

   3. 1/y as a function of 1/x in the case of the saturation-growth-rate model.

                                 In class: Show the linearization.

Since the data is now linearized, we can use a “linear best fit” to fit the transformed (linearized)
data. The slope and intercept of the transformed data are linked with the original models by:

   1. Slope=b and Intercept=ln(a) in the case of the exponential model.

   2. Slope=b and Intercept=ln(a) in the case of the power model.

   3. Slope=b/a and Intercept=1/a in the case of the saturation-growth-rate model.

From these relations, one can find the coefficients a and b in the case of all three models.
Example: In the case of drag .vs. wind velocity, it is physically intuitive that a good model
would be the power model. Indeed, the velocity should be zero when the wind velocity is zero.
Moreover, we do not expect any constraints or “leveling off” process (the faster the wind, the
greater the drag).

Solution: Plot ln(Drag) .vs. ln(Velocity). Use a linear fit. Extract the coefficients a and b of the
power model from the slope and intercept of the linear fit. This is described in figure 4.

                                   Figure 4: Using the power model.

From this we obtain that a=1.9842 and b=ln(-1.2941)=.2741. Therefore, the relationship between the
drag force and the wind velocity is:

                               F  .2741 V 1.9842

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