{
  "abstract": "Consider the linear regression of a response $Y$ on explanatory variables $X_1,\\dots,X_p$ with a positive definite joint correlation matrix. Let $\\mathbf{R}$ be the correlation matrix of the explanatory variables and $\\mathbf{r}$ the vector of their correlations with $Y$. The ellipsoid $E=\\{\\mathbf{x} : \\mathbf{x}^{\\top}\\mathbf{R}^{-1}\\mathbf{x}=1\\}$ is inscribed in the cube $[-1,1]^p$ and touches the face $x_j=1$ at $T_j$, the $j$th column of $\\mathbf{R}$. The point $P=\\mathbf{r}$ lies inside $E$, and three kinds of regression quantities are read off this diagram by elementary geometric constructions. The multiple correlation is $|OP|/|OP'|$, where $O$ is the origin and $P'$ is the point where the ray from $O$ through $P$ meets $E$ (for $P\\ne O$). The standardized regression coefficients are the coordinates of $P$ in the basis $T_1,\\dots,T_p$. The partial correlation between $X_i$ and $Y$ given the other explanatory variables is the position of $P$ on the chord of $E$ through $P$ parallel to the $x_i$-axis, on a linear scale from $-1$ to $1$ increasing with $x_i$; furthermore, the parallel chord through the origin has half-length $1/\\sqrt{\\mathrm{VIF}_i}$. Examples illustrate sign reversal and standardized coefficients outside the interval $[-1,1]$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.06536",
  "author_total": 1,
  "authors": [
    {
      "name": "Toshiyuki Shimono"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/epy2ten6409yy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.06536",
  "fulltext_url": null,
  "id": "epy2ten6409yy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.06536",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.ME"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2613|Advanced Statistical Methods and Models"
    ]
  ],
  "openalex": "W7220742411",
  "pdf_url": "https://arxiv.org/pdf/2610.06536",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/epy2ten6409yy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.06536",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.06536"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Reading Multiple Regression from an Ellipsoid Inscribed in the Unit Hypercube",
  "title_text": "Reading Multiple Regression from an Ellipsoid Inscribed in the Unit Hypercube",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:16Z",
  "url": "https://research.draconitic.com/paper/epy2ten6409yy",
  "venue": "arXiv",
  "versions": []
}