Nonlinear Data, Curve Fitting

• In many cases, data from experimental tests are not linear,
• so we need to fit to them some function other than a first-degree polynomial.
• Popular forms are the exponential form
  $$y=ax^b$$
  or
  $$y =ae^{bx}$$

• We can develop normal equations to the preceding development for a least-squares line by setting the partial derivatives equal to zero.
• Such nonlinear simultaneous equations are much more difficult to solve than linear equations.

• Thus, the exponential forms are usually linearized by taking logarithms before determining the parameters,

  For the case $y=ax^b$ $\Longrightarrow$

  $$ln y = ln a + b ln x$$

  For the case $y =ae^{bx}$ $\Longrightarrow$

  $$ln y = ln a + bx$$

• We now fit the new variable, $z= ln y$, as a linear function of $ln x$ or $x$ as described earlier (normal equations).
• Here we do not minimize the sum of squares of the deviations of $Y$ from the curve, but rather the deviations of $ln Y$.
• In effect, this amounts to minimizing the squares of the percentage errors, which itself may be a desirable feature.
• An added advantage of the linearized forms is that plots of the data on either log-log or semilog graph paper show at a glance whether these forms are suitable, by whether a straight line represents the data when so plotted.

• In cases when such linearization of the function is not desirable,
• or when no method of linearization can be discovered, graphical methods are frequently used;
• one plots the experimental values and sketches in a curve that seems to fit well.
• Transformation of the variables to give near linearity,
• such as by plotting against $1/x, 1/(ax + b), 1/x^2$,
• and other polynomial forms of the argument may give curves with gentle enough changes in slope to allow a smooth curve to be drawn.

• S-shaped curves are not easy to linearize; the relation
  $$y= ab^{c^x}$$
  is sometimes employed.
• The constants $a$, $b$, and $c$ are determined by special procedures.
• Another relation that fits data to an S-shaped curve is
  $$\frac{1}{y}= a + be^{-x}$$