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For observations $y_1,\dots ,y_n$ in a known covariate order, the isotonic least squares estimator solvesIt is the orthogonal projection of the observation vector onto a closed convex cone of nondecreasing sequences. Strict convexity of the squared distance gives a unique estimate.

For data $(3,1,2)$, the first pair violates monotonicity. Pooling it gives mean two, producing $(2,2,2)$ with squared error two. The pool adjacent violators algorithm repeatedly merges neighboring blocks whose means are out of order; the fitted value is constant at the block mean on each final block.

Positive observation weights replace squared error by ${\sum }_i{w}_i(y_i-{\theta }_i{)}^2$ and block means by weighted means. The order restriction regularizes the estimate without specifying a fixed polynomial form. The method requires the ordering to be meaningful; it does not discover an unknown covariate order from the response values.

The pool adjacent violators algorithm solves weighted isotonic regression: minimize ${\sum }_i{w}_i(y_i-{\theta }_i{)}^2$ with $w_i>0$ and ${\theta }_1\le \cdots \le {\theta }_n$. Start with singleton blocks. Whenever consecutive fitted block means decrease, merge the two blocks and replace them by their weighted mean; continue until no violation remains. For equal weights and observations $(3,1,2)$, merging the first two gives the solution $(2,2,2)$. A stack implementation revisits each block only when merging it, giving linear time. Pooling is required by the order-constrained optimum: two reversed adjacent blocks cannot remain separate fitted levels.