1D Recursive Filters

Compares a moving average, an exponential moving average, and a Kalman filter for smoothing a noisy 1D signal.

// VM is the shared utility library, loaded globally via _includes/head-scripts.html (js/**). This page uses VM.expressions.makeFunction, VM.expressions.makeNumber, VM.sampling.seededRandom, VM.sampling.gaussianRandom, VM.filters.movingAverageFilter, VM.filters.emaFilter, VM.filters.kalman1DFilter, VM.plotting.paddedRange, VM.ui.renderTable, VM.ui.applyExampleParams.
VM = window.VM

urlParams = new URLSearchParams(window.location.search)

methodOptions = ["True Signal", "Noisy measurements", "Simple Moving Average", "Exponential Moving Average", "Kalman Filter"]

methodSlugToLabel = new Map([
  ["true", "True Signal"],
  ["true-signal", "True Signal"],
  ["true_signal", "True Signal"],

  ["noisy", "Noisy measurements"],
  ["noisy-measurements", "Noisy measurements"],
  ["noisy_measurements", "Noisy measurements"],
  ["measurements", "Noisy measurements"],

  ["sma", "Simple Moving Average"],
  ["moving-average", "Simple Moving Average"],
  ["moving_average", "Simple Moving Average"],

  ["ema", "Exponential Moving Average"],
  ["low-pass", "Exponential Moving Average"],
  ["lowpass", "Exponential Moving Average"],
  ["exponential-moving-average", "Exponential Moving Average"],

  ["kalman", "Kalman Filter"],
  ["kalman-filter", "Kalman Filter"],
  ["kalman_filter", "Kalman Filter"]
])

methodLabelToSlug = new Map([
  ["True Signal", "true"],
  ["Noisy measurements", "noisy"],
  ["Simple Moving Average", "sma"],
  ["Exponential Moving Average", "ema"],
  ["Kalman Filter", "kalman"]
])

// Matches the line/marker colors mainPlot uses for each trace, so the
// overlay legend's swatches double as a color key. #9ca3af (gray) has no
// VM.plotting.colors() token -- left as a literal.
// vmTheme is what makes the colors below reactive. The chart's own chrome
// (background, axes, gridlines) repaints on a dark-mode toggle by itself,
// via the relayout patch in plotly-fullscreen-button.js -- but the *trace*
// colors are values this page bakes into the traces it builds, so nothing
// repaints them unless mainPlot re-runs. Depending on vmTheme is what
// re-runs it. See VM.plotting.onThemeChange for why it fires a couple of
// frames after the toggle rather than inline.
vmTheme = Generators.observe(notify => VM.plotting.onThemeChange(notify))

chartColors = VM.plotting.colors(vmTheme)

methodColors = new Map([
  ["True Signal", chartColors.fn],
  ["Noisy measurements", "#9ca3af"],
  ["Simple Moving Average", chartColors.warn],
  ["Exponential Moving Average", chartColors.ok],
  ["Kalman Filter", chartColors.alt]
])

initialMethods = {
  const raw = urlParams.get("show")
  if (raw === null || raw.trim() === "") return methodOptions

  const selected = raw
    .split(",")
    .map(s => s.trim().toLowerCase())
    .map(s => methodSlugToLabel.get(s))
    .filter(Boolean)

  return selected.length > 0 ? Array.from(new Set(selected)) : methodOptions
}

initialMax = urlParams.get("max") ?? "80"

initialAlpha = {
  const raw = Number.parseFloat(urlParams.get("alpha"))
  return Number.isFinite(raw) ? Math.min(Math.max(raw, 0), 0.99) : 0.9
}

initialK = {
  const raw = Number.parseInt(urlParams.get("k"), 10)
  return Number.isFinite(raw) ? Math.max(1, raw) : 8
}

initialR = {
  const raw = Number.parseFloat(urlParams.get("R"))
  return Number.isFinite(raw) ? Math.min(Math.max(raw, 0.001), 2) : 0.25
}

initialQ = {
  const raw = Number.parseFloat(urlParams.get("Q"))
  return Number.isFinite(raw) ? Math.min(Math.max(raw, 0), 1) : 0.05
}

// examples backs the "Try an example" dropdown near the top controls.
//
// Each params object lists "max" before "k" — windowK's range depends
// reactively on maxN, so its view gets recreated when max changes;
// VM.ui.applyExampleParams applies params in this order with a yield
// between each, so the freshly-recreated slider is what "k" ends up
// applied to. "show" holds the actual checkbox option labels (matching
// what Inputs.checkbox's view expects), not the URL's comma-separated
// slugs.
examples = [
  {
    title: "Sudden regime change",
    params: {f: "x > 10 ? 6 : 0", sigma: "0.5", b: "20", max: "80", alpha: 0.9, k: 8, R: 0.25, Q: 0.05}
  },
  {
    title: "Fast oscillation exposes lag",
    params: {f: "sin(3*x)", sigma: "0.3", b: "10", max: "100", alpha: 0.9, k: 6, R: 0.09, Q: 0.05}
  },
  {
    title: "Zero process noise stops tracking",
    params: {f: "sin(x)", sigma: "0.3", b: "20", max: "60", alpha: 0.9, k: 5, R: 0.09, Q: 0}
  },
  {
    title: "Overconfident sensor model",
    params: {f: "sin(x)", sigma: "0.6", b: "20", max: "60", alpha: 0.9, k: 5, R: 0.01, Q: 0.05}
  },
  {
    title: "Estimating a hidden constant",
    params: {f: "5", sigma: "1", b: "20", max: "60", alpha: 0.9, k: 8, R: 1, Q: 0}
  },
  {
    title: "Noisy sine wave",
    params: {f: "sin(x)", sigma: "0.4", b: "20", max: "60", alpha: 0.9, k: 5, R: 0.16, Q: 0.05}
  }
]

viewof fText = {
  const params = new URLSearchParams(window.location.search)
  const fTextInput = Inputs.text({label: "True signal f(x)", value: params.get("f") ?? "x > 10 ? 6 : 0", placeholder: "Example: sin(x), 5, cos(x/2)"})
  fTextInput.classList.add("ojs-fill")
  fTextInput.dataset.exampleField = "f"
  return fTextInput
}

// Plot commits the current signal/noise/endpoint fields and redraws the
// noisy measurements with a fresh seed — edits above don't take effect
// until this is clicked, so result/mainPlot don't recompute (and the
// noisy draw doesn't reshuffle) on every keystroke. The total-samples
// field, and the method legend and filter parameters overlaid on the
// chart, are unaffected and stay reactive.
viewof plotTrigger = {
  const plotButton = Inputs.button("Plot", {value: 0, reduce: v => v + 1})
  plotButton.classList.add("ojs-auto")
  return plotButton
}

viewof sigmaText = {
  const params = new URLSearchParams(window.location.search)
  const input = Inputs.text({label: "Gaussian Noise σ", value: params.get("sigma") ?? "0.5", placeholder: "Example: 0.4, 1/2"})
  input.dataset.exampleField = "sigma"
  return input
}

viewof bText = {
  const params = new URLSearchParams(window.location.search)
  const input = Inputs.text({label: "Right endpoint b", value: params.get("b") ?? "20", placeholder: "Example: 20, 4*pi"})
  input.dataset.exampleField = "b"
  return input
}

viewof maxNInput = {
  const input = Inputs.text({value: initialMax, label: "Total samples", placeholder: "Example: 60, 20*3"})
  input.dataset.exampleField = "max"
  return input
}

viewof exampleSelect = {
  const options = [null, ...examples]
  const select = Inputs.select(options, {label: "Try an example", format: e => e ? e.title : "Pick an example…", value: null})
  select.classList.add("ojs-fill")
  return select
}

// Plain CSS selectors, not direct viewof references -- see
// js/ui/apply-example.js's comment for why (windowK's view is reactively
// recreated whenever maxN changes, so referencing it by name here would
// make this cell -- and therefore applyExampleFromSelect -- reactive on
// it, forming a feedback loop with the fact that applying an example is
// what changes maxN in the first place).
exampleFieldSelectors = ({
  f: '[data-example-field="f"]',
  sigma: '[data-example-field="sigma"]',
  b: '[data-example-field="b"]',
  max: '[data-example-field="max"]',
  alpha: '[data-example-field="alpha"]',
  k: '[data-example-field="k"]',
  R: '[data-example-field="R"]',
  Q: '[data-example-field="Q"]'
})

// Applies the selected example's params to the fields above (including the
// filter parameter sliders overlaid on the chart below) and clicks Plot, in
// place -- no page reload, so no scroll jump either. A plain "change"
// listener attached once to the stable native <select> (rather than an OJS
// cell reactive on exampleSelect's value) -- a <select> fires both "input"
// and "change" per selection, and a reactive cell would run once per event,
// with two overlapping calls racing each other.
applyExampleFromSelect = {
  const nativeSelect = (viewof exampleSelect).querySelector("select")
  if (!nativeSelect) return
  nativeSelect.addEventListener("change", () => {
    const example = (viewof exampleSelect).value
    if (!example) return
    VM.ui.applyExampleParams(exampleFieldSelectors, example.params, viewof plotTrigger)
  })
}

// maxN parses the "Total samples" field's expression, falling back to 60
// for invalid input.
maxN = {
  const math = window.math
  const parsed = VM.expressions.makeNumber(math, maxNInput)
  if (parsed === null || !Number.isFinite(parsed)) return 60
  return Math.max(2, Math.round(parsed))
}

showTrueSignal = methods.includes("True Signal")

showNoisy = methods.includes("Noisy measurements")

showSMA = methods.includes("Simple Moving Average")

showEMA = methods.includes("Exponential Moving Average")

showKalman = methods.includes("Kalman Filter")

// Pressing Enter in the signal/noise/endpoint fields clicks Plot instead
// of submitting the field's own (invisible) form. Depends only on the
// stable views, so this wiring runs once and is never re-attached.
enterToPlot = {
  for (const view of [viewof fText, viewof sigmaText, viewof bText]) {
    const input = view.querySelector("input")
    if (!input) continue
    input.addEventListener("keydown", (e) => {
      if (e.key !== "Enter") return
      e.preventDefault()
      viewof plotTrigger.querySelector("button").click()
    })
  }
}

// Keeps the total sample count, filter parameters, and method choices
// shareable as these reactive controls change (independent of the Plot
// button).
urlSyncControls = {
  const params = new URLSearchParams(window.location.search)
  params.set("max", maxNInput)
  params.set("alpha", String(alpha))
  params.set("k", String(windowK))
  params.set("R", String(R))
  params.set("Q", String(Q))
  params.set("show", methods.map(m => methodLabelToSlug.get(m)).filter(Boolean).join(","))
  history.replaceState(null, "", `${window.location.pathname}?${params}`)
  return true
}

// committed snapshots fText/sigmaText/bText only when Plot is clicked, and
// writes everything back into the URL so the address bar stays a
// shareable link. It reads `viewof fText`/etc. (the stable DOM views)
// rather than `fText`/etc. (the reactive values), so it does NOT rerun on
// every keystroke — only when `plotTrigger` changes. It still runs once
// on initial page load, so the first render uses the URL-provided (or
// default) values with no click.
//
// The noisy draw's seed isn't a field the reader sets — it's derived from
// the clock (xored with plotTrigger's click count so two clicks in the
// same millisecond still differ), so every Plot click redraws fresh noise
// without exposing a seed control or cluttering the URL. Same pattern as
// apps/polynomial-fits/index.qmd's basePoints cell.
committed = {
  const math = window.math
  const fVal = (viewof fText).value
  const sigmaVal = (viewof sigmaText).value
  const bVal = (viewof bText).value

  const sigmaNum = VM.expressions.makeNumber(math, sigmaVal)
  const bNum = VM.expressions.makeNumber(math, bVal)
  const seedNum = Date.now() ^ (plotTrigger * 1000003)

  const params = new URLSearchParams(window.location.search)
  params.set("f", fVal)
  params.set("sigma", sigmaVal)
  params.set("b", bVal)
  params.set("max", maxNInput)
  params.set("alpha", String(alpha))
  params.set("k", String(windowK))
  params.set("R", String(R))
  params.set("Q", String(Q))
  params.set("show", methods.map(m => methodLabelToSlug.get(m)).filter(Boolean).join(","))
  history.replaceState(null, "", `${window.location.pathname}?${params}`)

  return {fText: fVal, sigma: sigmaNum, bEnd: bNum, seed: seedNum}
}

// result: for the current committed signal/noise/endpoint/seed and the
// live filter parameters (alpha, windowK, R, Q), draws maxN noisy
// measurements of f and runs all three filters over the entire stream.
//   rows — array of { i, x, trueVal, noisyVal, smaVal, emaVal, kalmanVal,
//     kalmanGain, smaErr, emaErr, kalmanErr }, one entry per sample
//     i = 0..maxN-1, x = i*b/(maxN-1).
//   f — parsed true signal, reused for the smooth reference curve in setup
result = {
  const math = window.math

  const f = VM.expressions.makeFunction(math, committed.fText)
  const fallback = x => NaN

  if (!f) return {rows: [], f: fallback}

  const sigma = committed.sigma
  const b = committed.bEnd
  const seed = committed.seed
  if (!Number.isFinite(sigma) || sigma < 0 || !Number.isFinite(b) || b <= 0 || !Number.isFinite(seed)) {
    return {rows: [], f}
  }

  const rng = VM.sampling.seededRandom(seed)
  const gaussian = VM.sampling.gaussianRandom(rng)

  const xs = []
  const trueVals = []
  const noisyVals = []
  for (let i = 0; i < maxN; i++) {
    const x = i * b / (maxN - 1)
    const trueVal = f(x)
    const noisyVal = trueVal + sigma * gaussian()
    xs.push(x)
    trueVals.push(trueVal)
    noisyVals.push(noisyVal)
  }

  const smaVals = VM.filters.movingAverageFilter(noisyVals, windowK)
  const emaVals = VM.filters.emaFilter(noisyVals, alpha)
  const kalman = VM.filters.kalman1DFilter(noisyVals, {R, Q})

  const rows = []
  for (let i = 0; i < maxN; i++) {
    rows.push({
      i, x: xs[i], trueVal: trueVals[i], noisyVal: noisyVals[i],
      smaVal: smaVals[i], emaVal: emaVals[i], kalmanVal: kalman.xs[i], kalmanGain: kalman.Ks[i],
      smaErr: Math.abs(smaVals[i] - trueVals[i]),
      emaErr: Math.abs(emaVals[i] - trueVals[i]),
      kalmanErr: Math.abs(kalman.xs[i] - trueVals[i])
    })
  }

  return {rows, f}
}

// setup computes the data needed by the main chart: a dense reference
// curve of the true signal over the entire committed domain, and axis
// ranges covering every sample.
setup = {
  if (result.rows.length === 0) {
    const empty = {lo: -3, hi: 3}
    return {refXs: [], refYs: [], xRange: empty, yRange: {lo: -1, hi: 1}}
  }

  const b = committed.bEnd
  const xRange = VM.plotting.paddedRange([0, b], {emptyRange: [-3, 3], relativePadding: 0.08, minPadding: 0.25})

  const sampleCount = 400
  const refXs = []
  const refYs = []
  for (let i = 0; i < sampleCount; i++) {
    const x = b * i / (sampleCount - 1)
    refXs.push(x)
    refYs.push(result.f(x))
  }

  const yValues = []
  for (const y of refYs) if (Number.isFinite(y)) yValues.push(y)
  for (const row of result.rows) {
    if (Number.isFinite(row.noisyVal)) yValues.push(row.noisyVal)
    if (Number.isFinite(row.smaVal)) yValues.push(row.smaVal)
    if (Number.isFinite(row.emaVal)) yValues.push(row.emaVal)
    if (Number.isFinite(row.kalmanVal)) yValues.push(row.kalmanVal)
  }
  const yRange = VM.plotting.paddedRange(yValues, {emptyRange: [-1, 1], relativePadding: 0.15, minPadding: 0.5})

  return {refXs, refYs, xRange, yRange}
}

// mainPlot draws the true signal (over the full domain, as a fixed
// reference, toggleable via showTrueSignal), every noisy measurement, and
// each enabled filter's estimate.
//
// _plotDiv is kept in a closure so Plotly reuses the same DOM node across
// reactive updates — this preserves zoom and pan when a parameter or
// checkbox changes.
mainPlot = {
  const Plotly = window.Plotly
  let _plotDiv = null

  return (rows, data, showTrueSignal, showNoisy, showSMA, showEMA, showKalman) => {
    const traces = []
    if (showTrueSignal) {
      traces.push({x: data.refXs, y: data.refYs, type: "scatter", mode: "lines", name: "True signal", showlegend: false, line: {color: chartColors.fn, width: 3}})
    }

    const xs = []
    const noisyYs = []
    for (const row of rows) {
      xs.push(row.x)
      noisyYs.push(row.noisyVal)
    }
    if (showNoisy) {
      traces.push({x: xs, y: noisyYs, type: "scatter", mode: "markers", name: "Noisy measurements", showlegend: false, marker: {color: "#9ca3af", size: 6, line: {color: chartColors.halo, width: 0.5}}})
    }

    if (showSMA && rows.length > 0) {
      const ys = []
      for (const row of rows) ys.push(row.smaVal)
      traces.push({x: xs, y: ys, type: "scatter", mode: "lines", name: "Simple Moving Average", showlegend: false, line: {color: chartColors.warn, width: 2.5}})
    }
    if (showEMA && rows.length > 0) {
      const ys = []
      for (const row of rows) ys.push(row.emaVal)
      traces.push({x: xs, y: ys, type: "scatter", mode: "lines", name: "Exponential Moving Average", showlegend: false, line: {color: chartColors.ok, width: 2.5}})
    }
    if (showKalman && rows.length > 0) {
      const ys = []
      for (const row of rows) ys.push(row.kalmanVal)
      traces.push({x: xs, y: ys, type: "scatter", mode: "lines+markers", name: "Kalman Filter", showlegend: false, line: {color: chartColors.alt, width: 2.5}, marker: {color: chartColors.alt, size: 5}})
    }

    const layout = {
      xaxis: {title: "x", range: [data.xRange.lo, data.xRange.hi], zeroline: true},
      yaxis: {title: "value", range: [data.yRange.lo, data.yRange.hi], zeroline: true},
      hovermode: "closest",
      uirevision: "static",  // keeps zoom/pan state across reactive updates
      autosize: true
    }
    const config = VM.plotting.config()

    // First render: create the div. Later renders: update in place (preserves zoom).
    if (!_plotDiv) {
      _plotDiv = document.createElement("div")
      _plotDiv.className = "plotly-box-large"
      Plotly.newPlot(_plotDiv, traces, layout, config)
      // newPlot fires before the div is in the DOM, so Plotly measures 0 width.
      // A ResizeObserver catches the real size once it's inserted and laid out,
      // and keeps correcting it on any later layout/viewport change.
      VM.plotting.autoResize(_plotDiv)
    } else {
      Plotly.react(_plotDiv, traces, layout, config)
    }
    return _plotDiv
  }
}

mainPlot(result.rows, setup, showTrueSignal, showNoisy, showSMA, showEMA, showKalman)

viewof methods = {
  const items = []
  for (const label of methodOptions) items.push({label, color: methodColors.get(label)})
  const legend = VM.ui.legendOverlay(items, {value: initialMethods})
  legend.dataset.exampleField = "show"
  return legend
}

viewof alpha = {
  const input = Inputs.range([0, 0.99], {step: 0.01, value: initialAlpha, label: "EMA smoothing α"})
  input.dataset.exampleField = "alpha"
  return input
}

viewof windowK = {
  const upper = Math.max(1, Math.min(50, maxN))
  const input = Inputs.range([1, upper], {step: 1, value: Math.min(initialK, upper), label: "SMA window k"})
  input.dataset.exampleField = "k"
  return input
}

viewof R = {
  const input = Inputs.range([0.001, 2], {step: 0.001, value: initialR, label: "Kalman R"})
  input.dataset.exampleField = "R"
  return input
}

viewof Q = {
  const input = Inputs.range([0, 1], {step: 0.001, value: initialQ, label: "Kalman Q"})
  input.dataset.exampleField = "Q"
  return input
}

NoteConvergence plots

// iteratesPlot compares each filter's "weight on the newest sample" —
// the fraction of the new estimate that comes directly from the latest
// measurement rather than the filter's own history. For EMA this weight
// is the fixed constant 1-α; for SMA it's the fixed constant 1/k; for
// Kalman it's the gain K_i, which the filter adapts on its own every
// step.
iteratesPlot = {
  const kalmanName = "Kalman gain (adaptive)"
  const emaName = "EMA weight 1-α (fixed)"
  const smaName = "SMA weight 1/k (fixed)"

  const marks = []
  const colorDomain = []
  const colorRange = []

  if (showKalman) {
    const pts = []
    for (const row of result.rows) pts.push({x: row.i, y: row.kalmanGain})
    marks.push(Plot.lineY(pts, {x: "x", y: "y", stroke: () => kalmanName}))
    colorDomain.push(kalmanName)
    colorRange.push(chartColors.alt)
  }

  if (showEMA) {
    marks.push(Plot.ruleY([1 - alpha], {stroke: () => emaName, strokeDasharray: "4,3"}))
    colorDomain.push(emaName)
    colorRange.push(chartColors.ok)
  }

  if (showSMA) {
    marks.push(Plot.ruleY([1 / windowK], {stroke: () => smaName, strokeDasharray: "4,3"}))
    colorDomain.push(smaName)
    colorRange.push(chartColors.warn)
  }

  const chart = Plot.plot({
    width: 340, height: 300, marginLeft: 46,
    x: {label: "sample index"},
    y: {label: "weight on newest sample", domain: [0, 1]},
    color: {domain: colorDomain, range: colorRange},
    marks
  })
  const legend = Plot.legend({color: {domain: colorDomain, range: colorRange}})

  const container = document.createElement("div")
  container.append(chart, legend)
  return container
}
// convergencePlot draws each enabled filter's running RMSE (root-mean-
// square error against the true signal, accumulated from sample 0 up to
// sample i) vs sample index, for the entire run.
convergencePlot = {
  const smaName = "SMA running RMSE"
  const emaName = "EMA running RMSE"
  const kalmanName = "Kalman running RMSE"

  const runningRmse = (errKey) => {
    const pts = []
    let sumSquares = 0
    for (let i = 0; i < result.rows.length; i++) {
      const err = result.rows[i][errKey]
      sumSquares += err * err
      pts.push({x: result.rows[i].i, y: Math.sqrt(sumSquares / (i + 1))})
    }
    return pts
  }

  const marks = []
  const colorDomain = []
  const colorRange = []

  if (showSMA) {
    marks.push(Plot.lineY(runningRmse("smaErr"), {x: "x", y: "y", stroke: () => smaName}))
    colorDomain.push(smaName)
    colorRange.push(chartColors.warn)
  }
  if (showEMA) {
    marks.push(Plot.lineY(runningRmse("emaErr"), {x: "x", y: "y", stroke: () => emaName}))
    colorDomain.push(emaName)
    colorRange.push(chartColors.ok)
  }
  if (showKalman) {
    marks.push(Plot.lineY(runningRmse("kalmanErr"), {x: "x", y: "y", stroke: () => kalmanName}))
    colorDomain.push(kalmanName)
    colorRange.push(chartColors.alt)
  }

  const chart = Plot.plot({
    width: 340, height: 300, marginLeft: 46,
    x: {label: "sample index"},
    y: {label: "running RMSE"},
    color: {domain: colorDomain, range: colorRange},
    marks
  })
  const legend = Plot.legend({color: {domain: colorDomain, range: colorRange}})

  const container = document.createElement("div")
  container.append(chart, legend)
  return container
}

NoteSample table

// Sample table — shows every sample in the run, with every filter's
// estimate, the Kalman gain, and each method's error against the true
// signal.
sampleTable = {
  if (result.rows.length === 0) {
    // Always return a Node so Quarto's OJS runtime never latches this
    // declaration cell as hidden.
    return document.createElement("div")
  }

  const formattedRows = []
  for (const row of result.rows) {
    formattedRows.push([
      row.i,
      row.x.toFixed(3),
      row.trueVal.toPrecision(6),
      row.noisyVal.toPrecision(6),
      row.smaVal.toPrecision(6),
      row.emaVal.toPrecision(6),
      row.kalmanVal.toPrecision(6),
      row.kalmanGain.toFixed(4),
      row.smaErr.toExponential(3),
      row.emaErr.toExponential(3),
      row.kalmanErr.toExponential(3)
    ])
  }

  return VM.ui.renderTable({
    html,
    headers: ["i", "x", tex`f(x)`, tex`z_i`, "SMA", "EMA", "Kalman", tex`K_i`, "|SMA err|", "|EMA err|", "|Kalman err|"],
    csvHeaders: ["i", "x", "true f(x)", "noisy z_i", "SMA", "EMA", "Kalman", "Kalman gain K_i", "SMA abs error", "EMA abs error", "Kalman abs error"],
    filename: "1d-recursive-filters-samples.csv",
    rows: formattedRows
  })
}

Each of these filters updates a running estimate from only the previous estimate and the newest measurement \(z_i\) — no stored history of the stream — which is what makes them run in O(1) memory and O(1) compute per sample. The simple moving average and exponential moving average (a.k.a. a first-order low-pass filter) both trade responsiveness for smoothness through a hand-picked constant (\(k\) or \(\alpha\)) that never changes:

\[ \text{SMA: } \hat x_i = \frac{1}{k}\sum_{j=i-k+1}^{i} z_j, \qquad \text{EMA: } \hat x_i = \alpha\, \hat x_{i-1} + (1-\alpha)\, z_i. \]

The Kalman filter instead maintains a running estimate of its own uncertainty \(P_i\) and derives its gain from it at every step, given only two beliefs about the noise: \(R\), the assumed measurement-noise (sensor) variance, and \(Q\), the assumed process-noise variance (how much the true value is expected to drift between samples):

\[ P_i^- = P_{i-1} + Q,\qquad K_i = \frac{P_i^-}{P_i^- + R},\qquad \hat x_i = \hat x_{i-1} + K_i\,(z_i - \hat x_{i-1}),\qquad P_i = (1-K_i)\,P_i^-, \]

with no \(\alpha\) or \(k\) to tune by hand — open the convergence plots on the constant-signal example below to watch \(K_i\) shrink automatically as the filter grows more confident.

References

Staszewski, Kuba. “Recursive Filters.” Blog post. https://www.staszewski.xyz/blog/recursive-filters/