Where does elimination collide?

Gaussian elimination is a systematic procedure for solving a system of linear equations by using row operations to remove one unknown at a time until the system is triangular and can be solved by back-substitution. Gauss described the method in 1810 while computing the orbit of the asteroid Pallas from redundant observations, applying it to the normal equations of least squares. Grcar's 2011 historical review Mathematicians of Gaussian elimination traced the algorithm and noted that equivalent elimination of unknowns was used long before Gauss, by Chinese mathematicians and by Newton, with Gauss standardising the notation. The referent here is an algorithm on numbers arranged in a matrix, counted in arithmetic operations, and it has nothing to do with removing atoms from a molecule, dropping predictors from a statistical model, or discarding options in a choice.[1][2]

Unit

One run of the algorithm on a coefficient matrix, scaled by its number of rows and columns

Recovery means

Not applicable

Measured by

Assessed by operation count (about two-thirds of n-cubed for an n by n system) and by numerical stability under pivoting

An elimination reaction removes two atoms or groups from adjacent carbons of a molecule and forms a new double bond in their place, the reverse of an addition. Hofmann reported the first clear case in 1851, heating a quaternary ammonium hydroxide so that it lost an amine and a proton to give an alkene, in the transformation now called Hofmann elimination. Saytzeff's 1875 study of the order of addition and loss of hydrogen iodide in organic compounds established the rule that the more substituted alkene usually predominates. The referent is a chemical transformation of covalent bonds, tracked by product yield and rate, and it is not an algorithm on a matrix, a model-building step in statistics, or a rule for choosing among options.[3][4]

Unit

One conversion of a substrate into an unsaturated product by loss of a leaving group and an adjacent atom

Recovery means

Not applicable

Measured by

Followed by reaction rate, product distribution, and the ratio of the more to the less substituted alkene

Backward elimination is a variable-selection procedure that begins with every candidate predictor in a regression model and then repeatedly deletes the least useful one until a stopping rule is met. Efroymson set out the computational stepwise version in 1960 for early digital computers, and Draper and Smith's Applied Regression Analysis gave the standard textbook treatment of the forward, backward, and stepwise variants. The referent is a model-building step that discards columns of data, judged by significance tests or information criteria, and it is unrelated to solving a linear system, breaking a chemical bond, or ruling out choices.[5][6]

Unit

One deletion step that drops a single predictor from the current model

Recovery means

Not applicable

Measured by

Guided by partial F tests, p-value thresholds, or information criteria such as AIC at each step

Elimination by aspects is a model of how people choose among options by attributes rather than by an overall score: at each stage the chooser picks an aspect, often weighted by its importance, and discards every option that lacks it, until one remains. Tversky introduced the theory in 1972 in Elimination by aspects: A theory of choice, showing that it can violate the principle that adding an option should not change the relative ranking of the others. The referent is a psychological decision rule that eliminates alternatives, studied through observed choices, and it does not describe row reduction, a chemical bond change, or the pruning of a regression model.[7]

Unit

One choice among a set of options, described by the aspects used to screen them

Recovery means

Not applicable

Measured by

Estimated from choice probabilities across option sets and tested against the constant-ratio and regularity conditions