Modeling with Power Functions

Objectives

1. Use plotPoints() to obtain a log-log plot of data.
2. Use a log-log plot to decide whether a power model is appropriate.
3. Given data exhibiting a power relationship, approximate the parameters by hand.
4. Use log rules to show a power function becomes $\ln(y) = m\ln(x) + b$ in log-log form, where $k = m$ and $C = \exp(b)$.
5. In R, use fitModel() to obtain the best-fit power model.
6. Evaluate a power model at a given input and interpret the answer.

1. Log-Log Plots

Objective. Use plotPoints() to obtain a log-log plot of data.

The data point $(0, 5)$ does not appear on a log-log plot, because $\ln(0)$ is undefined. The definition requires $x_0 > 0$.

The semi-log plot is defined on the Lesson 5 sheet, section 4. In R, log() is the natural log $\ln$. The log-log plot of a dataset D with input x and output y comes from

For data typed into R as vectors, put the vector names in the formula and leave off the data = argument of plotPoints() or fitModel(), as in Example 2.

Zero and negative values. The natural log takes positive numbers only. R returns -Inf for the log of 0 and NaN (“not a number”) for the log of a negative number:

> log(0)
[1] -Inf
> log(-2)
[1] NaN
Warning message:
In log(-2) : NaNs produced

Every input and every output must be positive to appear on a log-log plot, because the logarithm of zero or of a negative number is undefined. Modeling data with a zero or a negative value takes a method beyond this course.

Next: 2. Reasonable Power ModelsLesson map

2. Reasonable Power Models

Objective. Use a log-log plot to decide whether a power model is appropriate.

The function $y = 3 \cdot 2^{x}$ is not a power function, because its input $x$ is in the exponent. The definition puts the input in the base, raised to a constant exponent $k$.

On this sheet the domain of a power function is $x > 0$, where $x^{k}$ is defined for every $k$ and $\ln(x)$ exists.

A linear model $y = mx + b$ with $b \neq 0$ fitted to a dataset is not a power model, because $mx + b$ is then not of the form $Cx^{k}$. The definition requires a power function.

In a power function the input is the base and the exponent is fixed. In the exponential model $y = Ce^{k(x-h)}$ the input sits in the exponent. A power function has exactly two parameters, $C$ and $k$. The horizontal shift $h$ belongs to the exponential model alone (Lesson 6 sheet, section 3).

For positive $x$, the graphs of $y = e^{x}$ and $y = x^{3}$ both rise and bend upward. A curved standard plot therefore cannot separate the two families. The logged plots separate them. The semi-log plot of $y = e^{x}$ is the line $\ln(y) = x$, and the log-log plot of $y = x^{3}$ is the line $\ln(y) = 3\ln(x)$. The semi-log plot of $y = x^{3}$ and the log-log plot of $y = e^{x}$ are curves.

Build all three plots and compare them. Two of the three plots can be nonconstant linear at once.

Horizontal log-log line. A horizontal log-log line $\ln(y) = b$, with slope 0, gives the constant $y = e^{b}$. A power function has a nonzero exponent $k$, and the criterion therefore asks for a nonconstant log-log line.

Equally straight plots. When two of the plots look equally straight, both models are reasonable by the graphical test. The choice between them then rests on what is known about how the quantity grows: a quantity known to double over a fixed span of input calls for an exponential model.

Next: 3. Parameters by HandLesson map

3. Parameters by Hand

Objective. Given data exhibiting a power relationship, approximate the parameters by hand.

Fitting a power model by hand.

1. Take the data points with the smallest and the largest input, $(x_1, y_1)$ and $(x_2, y_2)$.

2. Find the slope from the logs of both coordinates:

   $$m = \frac{\ln(y_2) - \ln(y_1)}{\ln(x_2) - \ln(x_1)}.$$

3. Find the intercept by substituting the first point into $\ln(y) = m\ln(x) + b$:

   $$b = \ln(y_1) - m\ln(x_1).$$
4. Convert with $k = m$ and $C = e^{b}$ (the log-log form theorem, section 4), giving $y = Cx^{k}$.

For a dataset D with input x, the commands D[which.min(D$x), ] and D[which.max(D$x), ] print the two rows of step 1. To find them in the table instead, type View(D) in the Console and click the input’s column heading to sort by input.

Intercept by substitution. The Lesson 6 shortcut $b = \ln(y_1)$ (Lesson 6 sheet, section 2) works for the shifted semi-log line, because that line takes the value $b$ at the first input, where $x - h = 0$. A log-log line takes the value $b$ where $\ln(x) = 0$, that is, at $x = 1$, and the smallest input of a dataset is rarely 1.

Next: 4. Power Functions in Log-Log FormLesson map

4. Power Functions in Log-Log Form

Objective. Use log rules to show a power function becomes $\ln(y) = m\ln(x) + b$ in log-log form, where $k = m$ and $C = \exp(b)$.

The left graph shows a power function with $C > 0$, and the right graph is its log-log plot. Each point $(x_0, y_0)$ on the curve moves to $(\ln(x_0), \ln(y_0))$. The curve becomes a line with slope $m = k$ that crosses the vertical axis, where $\ln(x) = 0$, at $b = \ln(C)$.

The power function $y = -2x^{3}$ has no log-log form, because $C = -2$ makes every output with $x > 0$ negative, and $\ln(y)$ is undefined for a negative $y$. The definition requires $C > 0$.

For $x > 0$ and $C > 0$, taking the natural log of both sides of $y = Cx^{k}$ and applying two log rules (Lesson 5 sheet, section 1) gives a line:

Next: 5. Best-Fit Power Model in RLesson map

5. Best-Fit Power Model in R

Objective. In R, use fitModel() to obtain the best-fit power model.

The by-hand fit uses two points, and fitModel() uses every point. The formula given to fitModel() is the log-log line, and coef() prints that line’s $m$ and $b$.

Fitting a power model in R.

Model = fitModel(log(y) ~ m*log(x) + b, data = D)
coef(Model)

Storing the model with = prints nothing. The command coef(Model) prints $m$ and $b$ of the log-log line. Convert $m$ and $b$ to $k$ and $C$ with the log-log form theorem.

Fitted model function. The stored Model is the fitted model function (Lesson 6 sheet, section 4). With a log-log formula, Model(x) returns the power model’s output $Cx^{k}$.

Next: 6. Predictions and InterpretationLesson map

6. Predictions and Interpretation

Objective. Evaluate a power model at a given input and interpret the answer.

Evaluate the fitted model function by name, as in RPMModel(300). R then carries every digit of $C$ and $k$.

The input goes in the data’s own units. When a question gives the input in meters and the data record centimeters, convert the input value to centimeters before the call.

The interpretation is one sentence, “The model predicts that …”, with the units of the input and the output written out. Interpolation and extrapolation are defined on the Lesson 4 sheet, section 3.

The error is the observed output at a data point minus the model’s output at that point’s input.

A positive error means the model predicts too low at that point. A negative error means the model predicts too high.

Lesson map