Section 8.6 fits four rows exactly with a cubic and calls the zero residual memorisation; section 8.5 shows that dependent columns leave a family of equally good fits and that the shortest member is a convention. Every column you add lowers the training residual and gives the coefficients one more way to trade against one another. The field has no procedure that guarantees the right number of columns for every data source: holding rows back, penalising large coefficients, and stopping when the residual stops falling each rest on assumptions that can fail on another table.
Take a position on this question: what procedure should choose the number of columns, and what should a reported residual be allowed to promise?
In your answer: state how you divide the rows and what each part is used for; say how you would detect that an added column is memorising rather than explaining; say what the reported number can and cannot promise about rows nobody has recorded; and name one way your own procedure could mislead you even when every step is followed.
Rubric. Tick each criterion your answer meets, then save the self-assessment. The platform does not grade it.
Worked solution
There is no single correct answer, so this is a model of the shape a complete answer takes, not a key. A level-3 answer fixes three parts of the table before computing anything. The fitting part carries the normal equations and produces the coefficients for each candidate number of columns. The selection part is scored once for each candidate, and the candidate with the smallest selection residual is kept; ties go to the smaller number of columns, stated in advance. The test part is read once, after the choice is fixed, and the reported number is the residual sum of squares on it, written with its denominator, such as 12.4 over 40 rows. The answer then says what that number promises: it estimates the residual on further rows drawn like the test rows, from the same instrument, the same range of times and the same conditions, and it does not promise the same residual for rows from another instrument or another range. It names a failure: if rows arrive in groups, a random split can put rows from the same group on both sides, and the test residual falls while the model has not improved. It also says that a selection part of six rows cannot separate a line from a cubic whose true residuals differ by 0.1, so the choice of the number of columns is itself noisy. A strong answer may add a penalty on the coefficient length and say that this replaces the hard choice with a deliberate bias, at the cost of shrinking a real coefficient. The answer that scores 0 says only that a cubic can fit four points and stops there.