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The hat matrixis the orthogonal projection onto the column space of a full-rank design matrix, and the fitted response is $\hat{Y}=HY$.

In R, lm(y ~ 1, data=d) fits an intercept-only linear model, lm(y ~ x1 + x2, data=d) adds quantitative predictors, and a factor predictor is expanded into indicator columns.

Linear regression models a response's conditional expectation as a linear combination of specified predictors. In matrix form,where the design matrix $X$ describes the observations, $\beta$ contains the regression coefficients, and $\epsilon$ contains the errors. The fitted values $X\hat{\beta }$ estimate the response means. The model is linear in its regression coefficients; its columns may include powers, interactions, or other transformed predictors, so polynomial regression can fit curved relationships using the same machinery.

For a concrete example, take five constructed observationsAn ordinary least squares fit with an intercept gives the line $\hat{y}=0.8+x$. Its regression residuals are $(0.2,0.2,-0.8,0.2,0.2)$ and its residual sum of squares is $0.8$. A one-unit increase in the predictor corresponds to a one-unit increase in the fitted response. This is a description of this fitted association; interpreting a change as a causal effect requires the assumptions of a causal inference problem.

The geometric reason for the fit is simple. The possible mean vectors $X\beta$ form the column space of $X$. Ordinary least squares chooses the orthogonal projection of the data vector onto that vector subspace, so the remaining regression residual is orthogonal to every permitted fitted direction. The least-squares normal equations express this orthogonality as $X^T(Y-X\hat{\beta })=0$. The data's explained component and unexplained component are therefore separated by the same geometry used in a Hilbert space.

The optimization definition makes sense for any observed data. Quantifying the uncertainty of the regression coefficients needs additional assumptions about the sampling and errors. With fixed $X$ of full column rank and mean-zero errors with covariance matrix ${\sigma }^{2}I$, ordinary least squares is unbiased and has covariance ${\sigma }^2(X^{T}X{)}^{-1}$; the Gauss-Markov theorem compares it with other linear unbiased estimators. A normal linear model additionally makes Student t-distribution intervals and tests exact in finite samples. Model adequacy and predictive assessment remain relevant even when these calculations are available.

Two historical settings help explain the subject's appeal. Adrien-Marie Legendre's Nouvelles méthodes pour la détermination des orbites des comètes (1805) includes an appendix on the method of least squares: observations give more equations than unknowns, and squared errors provide a criterion for reconciling them. Its astronomical setting shows the motivation for combining imperfect measurements. ^[ref] Francis Galton's Regression Towards Mediocrity in Hereditary Stature (1886) examines the association between parental and offspring heights and the tendency of group averages toward the population average. This is a historical use of the regression name; the modern linear model is a general statistical formulation. ^[ref]

Orthogonal projections of an isotropic Gaussian vector are independent. If $P$ is an orthogonal projection of rank $r$ and $Z\sim N(0,I)$, then

Nonparametric regression estimates a regression function without restricting it to a fixed finite-dimensional family.

Density estimation constructs an estimate of an unknown probability density function from sampled observations.

This local phase change is a gauge transformation of the wave function. Multiplication by $e^{ia\psi (x)}$ is a gauge transform that preserves pointwise modulus and mass while shifting the gradient by $ia(\nabla \psi )u$.

For the mass-critical focusing nonlinear Schrödinger equation in dimension $d$, the NLS ground state is the positive radial solution ofIts symmetry orbit consists of the optimizers of the sharp Gagliardo-Nirenberg inequality.

For a radial function $u\in H^1({\mathbb{R}}^d)$,This supplies spatial decay without requiring a weighted moment.

A localized virial identity differentiates a weighted mass $\int \chi |u{|}^2$ and its associated momentum. Compactly supported or flattened weights retain the coercive interior contribution while replacing an infinite-variance assumption by controllable tail errors.

An $H^1$ solution of the nonlinear Schrödinger equation either exists globally forward in time or its $H^1$ norm, equivalently its gradient norm when mass is conserved, becomes unbounded at the finite endpoint of its maximal lifespan.

For the focusing power equation $i{u}_t+\Delta u+|u{|}^{p-1}u=0$, the conserved energy is

For sufficiently regular solutions of a gauge-invariant nonlinear Schrödinger equation,is independent of time.

In the convention used here, a Brzozowski NFA has one final state, every state can reach it, and each word labels a path to it from exactly one starting state.

The powerset construction turns an NFA with state set $Q$ into a DFA with state set $\mathcal{P}(Q)$, transitionand accepting subsets that meet the NFA final set.

Without epsilon transitions, defineA word is accepted when the set reached from the initial state meets the final set.

In $R=\mathbb{Z}[\sqrt{-5}]$, the idealhas additive index two. If $I=(a+b\sqrt{-5})$ were principal, multiplication by its generator would givewhich has no solution in integers. Thus $I$ is not principal.

The global phase symmetry of a Dirac field gives the conserved Dirac current $j^{\mu }=\bar{\psi }{\gamma }^{\mu }\psi$.

Spacetime translations give the canonical stress-energy tensor

For a point mass and $r>0$, writeThe vacuum scalar-curvature equation gives $r{f}^{\prime }+3f=0$, so $f=C/r^3$. The remaining vacuum equations fix $C=2GM$ in the corresponding Schwarzschild gauge.