25:["$","title",null,{"children":"Identify penguins by bill shape | Fil | Observable"}] 4b:T4aa,// Gaussian KDE, with R’s default bandwidth (nrd0); log variables are estimated on the log scale import * as science from "npm:science@2"; function kde(values: readonly number[], log = false): {x: number; y: number}[] { const v = values.filter((x) => x > 0).map(log ? Math.log : Number); const density = science.kde(v, {kernel: science.kernels.gaussian, bandwidth: science.bandwidths.nrd0}); const [lo, hi] = d3.extent(v); return d3.ticks(lo - 2 * density.bandwidth, hi + 2 * density.bandwidth, 100) .map((t) => ({x: log ? Math.exp(t) : t, y: density(t)})); } const densityVars = [ {key: "length", label: "Bill length L (mm)", value: (d) => d.culmen_length_mm}, {key: "depth", label: "Bill depth D (mm)", value: (d) => d.culmen_depth_mm}, {key: "size", label: "Size = L × D (mm², log)", value: (d) => d.culmen_length_mm * d.culmen_depth_mm, log: true}, {key: "shape", label: "Shape = L / D (log)", value: (d) => d.culmen_length_mm / d.culmen_depth_mm, log: true} ]; const densities = densityVars.flatMap(({key, value, log}) => d3.groups(penguins, (d) => d.species).flatMap(([species, rows]) => kde(rows.map(value), log).map((p) => ({...p, species, key})) ) );4c:T816,

4d:[] 47:{"id":"87cd24316143b7b0","slug":"penguin-bills-by-shape","title":"Identify penguins by bill shape","thumbnail":"2eb0c7e49078c5849a27a10fdee67611369df1d5cbbf9bc267f653e68750257c","access_level":"public","create_time":"$D2026-10-21T10:37:22.219Z","update_time":"$D2026-10-21T16:48:51.082Z","view_time":null,"first_public_time":"$D2026-10-21T11:12:38.713Z","has_unpublished_edits":null,"fork":false,"stars":4,"starred":false,"owner":{"type":"team","id":"da2a32154638c457","login":"fil","name":"Fil","avatar_url":"https://avatars-next.observableusercontent.com/d5b1777907117a2bdf2a07eb98600d7a9beb21639fde42b9753e95c88c7731a6","tier":"starter_2024","bio":"Vocateur.","following":false},"version":158,"publish_version":158,"role":null,"authored":false,"authors":[{"type":"individual","id":"45a379fcfcb14253","login":"fil","name":"Fil","avatar_url":"https://avatars-next.observableusercontent.com/d5b1777907117a2bdf2a07eb98600d7a9beb21639fde42b9753e95c88c7731a6","tier":null,"bio":"Vocateur.","following":false}],"members":[{"type":"individual","id":"45a379fcfcb14253","login":"fil","name":"Fil","avatar_url":"https://avatars-next.observableusercontent.com/d5b1777907117a2bdf2a07eb98600d7a9beb21639fde42b9753e95c88c7731a6","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":0,"collection":null,"type":"notebook","body":{"titling":"auto","title":"Identify penguins by bill shape","cells":[{"id":11,"value":"# Identify penguins by bill shape","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1,"value":"Following Michael Friendly’s [blog post](https://friendly.github.io/blog/posts/2026-10-peng-altviews/), we solve the penguins dataset's [Simpson's paradox](https://observablehq.com/@observablehq/plot-linear-regression-simpson) with a rotation of the coordinate system.\n\nBill length *L*, and depth *D*, can be expressed (without loss of information) as:\n\n- **Size** = *L* × *D* (bill profile area, mm²), and:\n- **Shape** = *L* / *D* (a dimensionless ratio, higher is thinner, lower is thicker)\n","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":12,"value":"const size = (d) => d.culmen_length_mm * d.culmen_depth_mm;\nconst shape = (d) => d.culmen_length_mm / d.culmen_depth_mm;","pinned":true,"hidden":false,"mode":"js","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":10,"value":"

\"Illustration — Bill length L and depth D. Artwork by Allison Horst, from palmerpenguins.

","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":8,"value":"---\n## Scatterplot in size–shape coordinates","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":13,"value":"This is in fact a 45° rotation of the (log *L*, log *D*) plane.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":5,"value":"Plot.plot({\n grid: true,\n color: {...color, legend: true},\n x: {type: \"log\", label: \"Size = L × D (mm²)\"},\n y: {type: \"log\", label: \"Shape = L / D\"},\n marks: [\n Plot.dot(penguins, {x: size, y: shape, stroke: \"species\", tip: true}),\n Plot.linearRegressionY(penguins, {x: size, y: shape, stroke: \"species\"}),\n Plot.linearRegressionY(penguins, {x: size, y: shape, strokeDasharray: \"2,8\", fillOpacity: 0.05})\n ]\n})","pinned":true,"hidden":false,"mode":"js","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":9,"value":"## The shape axis separates all three species\n\nKernel densities of the four bill variables by species. Length, depth, or size, all confuse two species. But on the shape axis, species barely overlap.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":7,"value":"

${\ndensityVars.map(({key, label, log}) =>\n Plot.plot({\n width: 300, height: 220,\n color,\n x: {label, type: log ? \"log\" : \"linear\", ticks: 5},\n y: {label: \"Density\", grid: true, ticks: 4},\n marks: [\n Plot.areaY(densities.filter((d) => d.key === key), {x: \"x\", y2: \"y\", fill: \"species\", fillOpacity: 0.35}),\n Plot.lineY(densities.filter((d) => d.key === key), {x: \"x\", y: \"y\", stroke: \"species\"}),\n Plot.ruleY([0])\n ]\n })\n)\n}

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