{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "5Y3DYMPVEG"
    },
    {
      "field": "about",
      "target": "6C67F22FTD"
    },
    {
      "field": "about",
      "target": "N3YMNMWANM"
    },
    {
      "field": "uses",
      "target": "6C67F22FTD"
    },
    {
      "field": "uses",
      "target": "KCVKGSYJR9"
    },
    {
      "field": "uses",
      "target": "KHA4WMRDKY"
    },
    {
      "field": "uses",
      "target": "MK4FDH3EDM"
    },
    {
      "field": "uses",
      "target": "N3YMNMWANM"
    },
    {
      "field": "uses",
      "target": "NA5X6JVK5J"
    },
    {
      "field": "uses",
      "target": "V6XGZQ0GR1"
    },
    {
      "field": "uses",
      "target": "VDEFF7A5Q4"
    },
    {
      "field": "uses",
      "target": "XK7W504YEW"
    },
    {
      "field": "uses",
      "target": "XKGEV5RP3T"
    }
  ],
  "body": "\n## Summary\n\nOrdinary least squares estimates the coefficients of a [[linear regression|41FVC4P5JB]] by minimising the sum of squared residuals.\nThe estimate has the closed form $\\hat\\beta = (X^T X)^{-1} X^T y$ when the design matrix has full column rank.\nGeometrically, the fitted values are the [[orthogonal projection|0B8KZRT28M]] of the data onto the column space of the design matrix.\nBy the [[Gauss–Markov theorem|N3YMNMWANM]], it is the best linear unbiased estimator under the classical assumptions.\n\n## Explanations\n\n### L1 human\n\nDraw a line through a cloud of points.\nFor each point, measure the vertical gap between the point and the line, and square it.\nOrdinary least squares picks the line that makes the sum of these squared gaps as small as possible.\nFor the points $(0, 0)$, $(1, 1)$ and $(2, 3)$, the best line is $y = -1/6 + 3x/2$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "62J05"
    ],
    "name": "Ordinary least squares",
    "same_as": {
      "wikidata": "Q2912993",
      "wikipedia": "Ordinary least squares"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Ordinary_least_squares"
      }
    ],
    "specializes": [
      "0BYTQ4X1HC"
    ],
    "tag": "YM708KHV2W"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/YM708KHV2W/",
  "links": [
    {
      "field": "specializes",
      "target": "0BYTQ4X1HC"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/YM708KHV2W.md",
  "msc": [
    "62J05"
  ],
  "name": "Ordinary least squares",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/YM7/YM708KHV2W.md",
  "status": null,
  "tag": "YM708KHV2W",
  "uri": "m2://YM708KHV2W",
  "url": "https://tamnd.github.io/math2/wiki/ordinary-least-squares/",
  "writing": {
    "tag": "YM708KHV2W",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}