designmatrisen

The designmatrisen, or design matrix, is a mathematical object used in statistical modeling to organize the predictor variables for a set of observations. It maps each observation to a row and each predictor to a column, so that the linear predictor β is obtained by multiplying the design matrix by the coefficient vector.

In a linear regression model y = Xβ + ε, X is the design matrix with n rows (observations) and p columns (predictors). Columns correspond to predictors; if an intercept is included, one column is a constant one. Categorical variables are encoded with dummy variables; continuous variables enter as raw values or transformed terms. The design matrix may also include polynomial, interaction, or spline terms to capture nonlinear effects.

Key formulas: the ordinary least squares estimate is β_hat = (X^T X)^{-1} X^T y, provided X^T X is invertible; in general β_hat = (X^T X)^{+} X^T y using a pseudoinverse. Predictions are y_hat = X β_hat.

Properties: the rank and conditioning of X affect identifiability and numerical stability. Multicollinearity (near-dependencies among columns) inflates variance of estimates. In practice, centering and scaling of predictors can improve numerical behavior; when using an intercept, one category of a categorical variable is typically omitted to avoid exact linear dependence.

Applications: design matrices are central to ordinary least squares, ANOVA, and generalized linear models; they appear in most statistical software and in experimental design analysis.

Example: for y = β0 + β1 x1 + β2 x2, with intercept, X = [[1, x11, x12], [1, x21, x22], ...].