Sigmoid Landmarks

The board shows $y = A\cdot\text{pnorm}(x, m, s) + v$ with the Lesson 9 landmarks for its parameters:

• the level on the left, $y = v$, and the level on the right, $y = v + A$;
• the whole rise $A$, negative when the curve falls;
• the halfway crossing $\left(m,\ v + \frac{A}{2}\right)$, where the curve is steepest;
• the five-sixths crossing, where the curve passes $v + \frac{5A}{6}$ at about $m + s$;
• the spread $s$, about the distance from $m$ out to the five-sixths crossing.

Every landmark is computed exactly from the curve. The five-sixths crossing is at $m + 0.967s$.

Under Data, TwitterUsers3 or NetherlandsPopulation is drawn under the curve for placing the landmarks by eye. Show numbers works the estimating recipe under Estimating steps with the current values. The estimates from Notes Example 4 and Example 6 are in the menu beside Model. Once the curve is moved off those estimates, the menu entry changes to “Changed values”. Whole view brings the whole curve and the data back into view.

Checking view steps through the check under Checking the estimate. The two levels are drawn first, then the halfway point and the five-sixths point. The S through the halfway and five-sixths points is drawn last.

Under More:

• 84 percent level. The level $v + 0.841A$ is marked. The curve passes it exactly at $m + s$, because $\text{pnorm}(1) = 0.841$.
• Two-thirds level. The level $v + \frac{2A}{3}$ is marked with the point where the curve crosses it.

Show R gives makeFun() and plotFun() code that draws the same curve in RStudio. When a dataset is chosen, the code has a plotPoints() line before the plotFun() line.

Keys: press L or R to pick a level. Press M or S to pick a crossing point. 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. In Checking view, the left and right arrow keys step through the check.