{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $p \u003e k \\ge 1$ be integers.\nThe *factor analysis model* with $k$ factors for a random vector $X$ in $\\R^p$ is\n$$X = \\mu + L F + \\varepsilon,$$\nwhere:\n\n1. $\\mu \\in \\R^p$ is a constant vector and $L$ is a constant $p \\times k$ matrix, the matrix of *loadings*.\n2. $F$ is a random vector in $\\R^k$, the *common factors*, with $\\mathbb{E}F = 0$ and covariance matrix $I_k$.\n3. $\\varepsilon$ is a random vector in $\\R^p$, the *specific errors*, with $\\mathbb{E}\\varepsilon = 0$ and diagonal covariance matrix $\\Psi$.\n4. $F$ and $\\varepsilon$ are uncorrelated.\n\nThe factors $F$ are not observed; only $X$ is.\n*Factor analysis* is the estimation of $L$ and $\\Psi$ from observations of $X$, and the interpretation of the factors.\n\n## Remarks\n\nThe model implies that the [[covariance matrix|0V7G6WMFK0]] of $X$ is $\\Sigma = L L^T + \\Psi$.\nExample: with $p = 3$, $k = 1$, $L = (1, 1, 1)^T$ and $\\Psi = I_3$, the covariance matrix has diagonal entries $2$ and off-diagonal entries $1$.\nNon-example: a model $X = \\mu + L F + \\varepsilon$ with $\\Psi$ not diagonal fails clause 3; the errors would then carry shared variation that the factors are meant to explain.\nThe loadings are determined only up to rotation: if $Q$ is a $k \\times k$ orthogonal matrix, then $L Q$ and $L$ give the same $\\Sigma$.\nFor this reason, rotations such as varimax are chosen to make the loadings easier to interpret.\nUnder normality, $L$ and $\\Psi$ are often estimated by maximum likelihood.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "4QQQP1Q886",
    "kind": "definition",
    "msc": [
      "62H25"
    ],
    "name": "Factor analysis",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Factor_analysis"
      }
    ],
    "tag": "2Y7P8SABM9",
    "uses": [
      "C63RVH96TF",
      "0V7G6WMFK0",
      "0WQNRCNY3M"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/2Y7P8SABM9/",
  "links": [
    {
      "field": "defines",
      "target": "4QQQP1Q886"
    },
    {
      "field": "uses",
      "target": "C63RVH96TF"
    },
    {
      "field": "uses",
      "target": "0V7G6WMFK0"
    },
    {
      "field": "uses",
      "target": "0WQNRCNY3M"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/2Y7P8SABM9.md",
  "msc": [
    "62H25"
  ],
  "name": "Factor analysis",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/2Y7/2Y7P8SABM9.md",
  "status": null,
  "tag": "2Y7P8SABM9",
  "uri": "m2://2Y7P8SABM9",
  "url": "https://tamnd.github.io/math2/wiki/factor-analysis-definition/",
  "writing": {
    "tag": "2Y7P8SABM9",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}