# Data Manipulation

### Fitting Data

Want process variable -> can't directly measure it?

DEFINITION:

**Calibration** involves establishing a
relationship between the process variable of interest and the
measured quantity

Once have C=f(x) -> plot data. *How do you use it?*

- Interpolation - get data
**between** existing points.
- Extrapolation - get data
**outside** existing points.
- Fit curve.

Interpolate:

$y = y_1 + \frac{x-x_1}{x_2-x_1}(y_2-y_1)$,

where x and y are the to-be-determined variables and x_i and y_i
are the data on the curve.

Need to do more often (or extrapolate) -> develop an
expression y=f(x)

### Curve Fitting

Straight line -> easy

- by inspection ("looks good")
- least squares analysis -> minimize the sum of (absolute
error)
^{2}

Eqn. of line -> *y = ax + b*

a - slope -> $\frac{rise}{run} = \frac{y_2-y_1}{x_2-x_1}$ b -
intercept -> $b = y_1-ax_1 = y_2-ax_2$

Non-linear fitting -> cast non-linear equation [like
*y=Ae*^{bx}] into linear form [like (*ln(y) = bx +
ln(a)*].

Plot one y-dependent quantity [here, ln(y)] versus an
x-dependent quantity [here, x].

OUTCOME:

Utilize curve-fitting or linear interpolation to
estimate unmeasured data from measured data

TEST YOURSELF!

How do you "linearize": *y=Bx*^{3}
[cheat]

These two types occur often:

- semi-log (log-linear or linear-log): useful for exponential
dependencies
- log-log: useful for polynomial dependencies

IMPORTANT:

Do *not* perform a "double log", that is,
don't plot the ln(y) on log paper.