A hat matrix is a human-made mathematical construct in linear regression analysis that maps observed response values to fitted values. It is an n×n projection matrix H = X(X'X)^{-1}X' where X is the design matrix, and its diagonal elements h_ii (leverage values) quantify how much each observation influences its own fitted value. The matrix persists through algebraic computation in statistical software and textbooks. [formal: matrix | substrate: mind | horizon: a moment | explicit: yes | epoch: 0.01]
Accepted ontology entry
hat matrix
A hat matrix is a human-made mathematical construct in linear regression analysis that maps observed response values to fitted values. It is an n×n projection matrix H = X(X'X)^{-1}X' where X is the design matrix, and its diagonal elements…
Definition
Why it is in scope
A square matrix used in linear regression that maps observed response values to fitted values (ŷ = Hŷ). It is a human-made mathematical construct, central to regression diagnostics because its diagonal elements (the leverages) quantify how much each observation influences its own fitted value.
Names and aliases
- hat matrixen · CANONICAL
Relations from this entry
- cms585syw015oi9iq1u1vwdy8INSTANCE_OF →
A hat matrix IS a specific kind of matrix — the projection matrix in linear regression that maps observed values to fitted values. A competent speaker would call a hat matrix 'a matrix.'
- cmsk6gz0j03yknobp6dik93doSERVES →
The hat matrix computes leverage (its diagonal) and is fundamental to computing studentized residuals, Cook's distance, and other regression diagnostics. Its designed purpose is to enable diagnostic plot computation.
- cmskmodwa04zunobpymq3i7ifSERVES →
The hat matrix is used for the sake of regression diagnostics — it computes leverages and influences that underpin diagnostic plots. Servant (hat matrix) → master (regression diagnostic). Its primary applied purpose in statistics is diagnostic analysis.
Relations to this entry
- cmskkckl404ujnobpjwzo6tr4← DERIVED_FROM
dfits is computed using the hat matrix (diagonal leverage values) and deleted-case residuals. The hat matrix existed first and fed into the construction of dfits as an influence measure. The which-came-first test: the hat matrix concept predates the dfits statistic.
- cmskst1fr05dhnobp30ptw49u← DEPENDS_ON
A leverage residual needs the hat matrix to operate: its formula r_i * sqrt((1-h_ii)/h_ii) requires h_ii (leverage) values from the hat matrix H=X(X'X)^{-1}X'. Remove the hat matrix and you cannot compute the leverage values needed to produce leverage residuals.
Record identity
- Created
- Aug 8, 2026, 11:45 AM UTC
- Content hash
- 6efef090f831f547d1165240ba4fea1450873ac0c3d88d9634bbc3a9d8bf0642