Sine Explorer

The model is the general sine form:

$$y = A\sin\!\left(\frac{2\pi}{P}(x - h)\right) + v$$

Explore has a slider for each of $A$, $P$, $h$ and $v$. The plot shows the center line $y = v$, the maximum $v + A$ and the minimum $v - A$. The amplitude $A$ and the period $P$, from one peak to the next, are marked with brackets. The horizontal shift $h$ is marked on the input axis at the first upward crossing at or to the right of 0. A slider value of $h$ outside 0 to $P$ is another upward crossing. For such a slider value, the plot shows the whole-period move from the slider’s crossing to the reported horizontal shift. $A$ is always positive.

Estimate is the by-hand procedure, applied to TemperaturesDanville or SunsetGreenwich. It takes five values, each clicked on the plot or typed:

• $y_{\max}$, a round level near the top of the dataset, and $y_{\min}$, a round level near the bottom;
• a peak and the next peak, or a trough and the next trough;
• the input of an upward crossing of the center line.

Show numbers gives the arithmetic on the values:

$$v = \frac{y_{\max} + y_{\min}}{2}, \qquad A = \frac{y_{\max} - y_{\min}}{2}$$

$P$ is the distance from one peak to the next, or from one trough to the next. With several cycles between the two peaks, the distance is divided by the cycle count. $h$ is the crossing, moved by whole periods until it lies between 0 and $P$. Once all five values are in, the curve is drawn on the data for comparison.

In Lesson 8 Example 4, the levels are 88 and 32. The peaks are at Months 7 and 55, with four cycles between the two peaks. The crossing is at Month 2.5. Show numbers gives $P = 48/4 = 12$. The model drawn is $28\sin\!\left(\frac{2\pi}{12}(\text{Month} - 2.5)\right) + 60$.

Under More:

• Upward crossings. Mark every crossing shows every translation of $h$ by whole periods. The curve is the same for every translation. The reported horizontal shift is the smallest of these values that is 0 or more.
• Kept curve. Keep this curve keeps a gray copy of the curve for comparison after a change.

Show R gives plotFun() code that plots the current curve in RStudio with the same values. When a dataset is shown, the code also has a plotPoints() line for the data.

Keys: in Explore, press A, P, H or V to pick a handle. Press an arrow key to move it one slider step. Hold Shift with an arrow key to move it five steps. Press Esc to let go.