9 · Your Phone Is a Sensor

Navigation apps estimate speed from the phones of a few drivers: see how many users they need, and what bias does

Navigation apps collect GPS “pings” from phones at regular intervals. A car that spends twice as long on a stretch sends twice as many pings, so averaging the pings gives a space-mean speed, just like the drone. The catch: only cars running the app are counted.

viewof pen = Inputs.range([1, 50], {value: 5, step: 1, label: "Cars running the app (%)"})
viewof appBias = Inputs.range([0, 20], {value: 0, step: 1, label: "App users drive faster by (%)"})
viewof probeRun = Inputs.button("Run 300 more days")

// 300 simulated five-minute periods of 180 vehicles each. The app's estimate is the harmonic mean
// of the app users' speeds (pings arrive in proportion to time spent on the road).
probe = {
  probeRun;
  const randn = () => Math.sqrt(-2 * Math.log(1 - Math.random())) * Math.cos(2 * Math.PI * Math.random());
  const rows = [];
  let none = 0;
  for (let r = 0; r < 300; r++) {
    let inv = 0, invApp = 0, nApp = 0;
    for (let i = 0; i < 180; i++) {
      const app = Math.random() < pen / 100;
      let v = Math.min(40, Math.max(8, 25 + 7 * randn()));
      if (app) v *= 1 + appBias / 100;
      inv += 1 / v;
      if (app) { invApp += 1 / v; nApp++; }
    }
    if (nApp === 0) { none++; continue; }
    rows.push({r, err: (nApp / invApp - 180 / inv) * 3.6});
  }
  const rmse = Math.sqrt(d3.mean(rows, d => d.err ** 2));
  return {rows, none, rmse, bias: d3.mean(rows, d => d.err)};
}

Plot.plot({
  width: plotW, height: 200,
  x: {label: "App estimate minus true space-mean speed (km/h)", domain: [-40, 40]},
  y: {axis: null},
  marks: [
    Plot.ruleX([0], {stroke: ink, strokeWidth: 2}),
    Plot.dot(probe.rows, Plot.dodgeY({x: "err", r: 2.4, fill: accent, fillOpacity: 0.6})),
  ]
})

md`Each dot is one five-minute period. Typical error: **±${probe.rmse.toFixed(1)} km/h** · average error (bias): **${probe.bias >= 0 ? "+" : ""}${probe.bias.toFixed(1)} km/h**${probe.none ? ` · in ${probe.none} of 300 periods, not a single app user passed` : ""}.`

What it shows

Two lessons: too few app users gives noise, and app users who drive differently give bias. More data cures noise, never bias. Lecture 2 places this among every other traffic sensor.

theme = Generators.observe(notify => {
  const read = () => notify(document.body.classList.contains("quarto-dark") ? "dark" : "light");
  read();
  const watcher = new MutationObserver(read);
  watcher.observe(document.body, {attributes: true, attributeFilter: ["class"]});
  return () => watcher.disconnect();
})
accent = theme === "dark" ? "#c9a0ff" : "#57068c"
accentRange = theme === "dark" ? ["#4b3a5c", "#ecdcff"] : ["#dccbee", "#3a0463"]
ink = theme === "dark" ? "#f2f2f2" : "black"
// Plot width: Plot's default 640 px, except on a phone, where a plot is drawn to fit its card
// instead of being shrunk (and its text with it). Changes only when the width actually changes.
plotW = Generators.observe(notify => {
  let last;
  const read = () => {
    const main = document.querySelector("main"), fit = main ? main.clientWidth - 50 : 640;
    const w = !document.querySelector(".reveal") && fit < 640 ? Math.max(240, Math.floor(fit)) : 640;
    if (w !== last) notify(last = w);
  };
  read();
  window.addEventListener("resize", read);
  return () => window.removeEventListener("resize", read);
})