Straightening X-Y Plots

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Community Portal → Data Analysis → Straightening X-Y Plots

1 Simple Functions
2 Not-So-Simple Functions
3 See Also
4 References

Given experimental data for $y$ measured at known values of $x$ , our problem is to identify the functional relation between the two. We invariably start by constructing an x-y plot on linear axes. Our goal is to find a simple relation between $x$ and $y$ ; that is, we seek a straight line. If our x-y plot is not straight, we try to straighten it by replotting, using other functions or other scales or both.^[1]

Simple Functions

Figure 1. An exponential curve linearizes by plotting on semilog axes.

Some functions readily linearize; examples include the exponential,

(1)

$y \ = \ A\, e^{mx}$

and the power law,

(2)

$y \ = \ \alpha \, x^{\beta}$

These linearize by taking logarithms, so (1) becomes

Figure 2. An power law curve linearizes by plotting on log-log axes.

(3)

$\ln y \ = \ \ln A + mx$

which gives a straight line on a semilog plot, as in Figure 1, while (2) becomes

(4)

$\ln y \ = \ \ln \alpha + \beta \ln x$

which gives a straight line on a log-log plot, as in Figure 2.

Equations (3) and (4) are examples of the kinds of transformations we seek, though in general we can expect the required functions to be more complicated than either exponentials or power laws.

Not-So-Simple Functions

If neither a semilog or log-log plot produces a straight line, then we need a systematic way to search for functions that will linearize our data. One way is to order functions of $x$ in a horizontal progression, as in Figure 3, and order those of $y$ in a vertical progression [10], also in Figure 3. We will use these progressions to guide our re-plotting of $y$ vs $x$ in our search for a linear relation.^[2]

Figure 3. Vertical progression of functions of $y$ and horizontal progression of functions of $x$
...
$y 3$
$y 2$
$y$
ln $y$
$y$ ^—1
$y$ ^—2
...	$x$ ^—2	$x$ ^—1	ln $x$	$x$	$x$ ²	$x$ ³	...

Typically, we start from a simple x-y plot on linear axes. Then, depending on the signs of the first and second derivatives, we move along the progressions in Figure 3 to find other functions of $x$ and $y$ that tend to straighten the curve. We identify four general possibilities, as shown by the Tukey Circle in Figure 4:

Figure 4. The Tukey Circle, on which each arc represents a possible shape of a monotonic, nonlinear curve obtained when

y

is plotted against

x

. To find new functions that tend to linearize the plot, we move along one of the progressions of functions shown in Figure 3. The directions of movement are indicated by the arrows. After Tukey ^[2].

If $d y / d x$ and $d 2 y / d x 2$ are both positive (bottom right quadrant in Figure 4), then move either to the right along the progression in $x$ or down along the progression in $y$ .
If both derivatives are negative (top right quadrant in Figure 4), then move either to the right along the progression in $x$ or up along the progression in $y$ .
If $d y / d x > 0$ while $d 2 y / d x 2 < 0$ (top left quadrant in Figure 4), then move either to the left along the progression in $x$ or up along the progression in $y$ .
If $d y / d x < 0$ while $d 2 y / d x 2 > 0$ (bottom left quadrant in Figure 4), then move either to the left along the progression in $x$ or down along the progression in $y$ .

Usually we must repeatedly apply the transformations in Figure 3 to find a (nearly) straight line. When this is necessary, it may help to let the current $f (y)$ become $y$ on Figure 3 and let the current $f (x)$ become $x$ .

Note that we may change the function of $x$ or that of $y$ or both. Some data will be more sensitive to changes in $x$ , while other data will be more sensitive to changes in $y$ . Often, the functions of both $x$ and $y$ must be changed to achieve a (nearly) straight line. The process for straightening curves is illustrated on an example page.

References

↑ J. M. Haile, Analysis of Data, Macatea Productions, Central, SC, 2003. ISBN 0-9728602-0-7.
↑ ^2.0 ^2.1 J. W. Tukey, Exploratory Data Analysis, Addison-Wesley, Reading, MA, 1977. ISBN 0201076160

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