Least Squares, Logistic, and Cox-Regression with Ordered
Predictors

Develops a classification model for ordinal responses.  A
logic regression trees is constructed for ordinal categories in
the data and then all trees are used to predict the class of
observations.

Method for develops a classification model for ordinal responses based on logic regression.

Compute least squares estimates of one bounded or two ordered
antitonic regression curves
Fadoua Balabdaoui, Kaspar Rufibach, Filippo Santambrogio
We consider the problem of estimating two antitonic
regression curves f1* and f2* under the constraint that f1* >=
f2*. Given two sets of n data points g1(x_1), ..., g1(x_n) and
g2(x_1), ..., g2(x_n) that are observed at (the same)
deterministic points x_1, ..., x_n, the estimates are obtained
by minimizing the Least Squares criterion L(f1, f2) =
sum_{i=1}^n (g1(x_i) - f1(x_i))^2 w1(x_i) + sum_{i=1}^n
(g2(x_i) - f2(x_i))^2 w2(x_i) over the class of pairs of
functions (f1, f2) such that f1 and f2 are antitonic and
f1(x_i) >= f2(x_i) for all i = 1, ..., n. The estimates are
computed with a projected subgradient algorithm where the
projection is calculated using a PAVA.

Compute least squares estimates of one bounded or two ordered
isotonic regression curves
We consider the problem of estimating two isotonic
regression curves g1* and g2* under the constraint that they
are ordered, i.e. g1* <= g2*. Given two sets of n data points
y_1, ..., y_n and z_1, ..., z_n that are observed at (the same)
deterministic design points x_1, ..., x_n, the estimates are
obtained by minimizing the Least Squares criterion L(a, b) =
sum_{i=1}^n (y_i - a_i)^2 w1(x_i) + sum_{i=1}^n (z_i - b_i)^2
w2(x_i) over the class of pairs of vectors (a, b) such that a
and b are isotonic and a_i <= b_i for all i = 1, ..., n. The
estimates are computed with a projected subgradient algorithm
where the projection is calculated using a PAVA.

We consider the problem of estimating two isotonic
regression curves g1* and g2* under the constraint that they
are ordered, i.e. g1* <= g2*. Given two sets of n data points
y_1, ..., y_n and z_1, ..., z_n that are observed at (the same)
deterministic design points x_1, ..., x_n, the estimates are
obtained by minimizing the Least Squares criterion L(a, b) =
sum_{i=1}^n (y_i - a_i)^2 w1(x_i) + sum_{i=1}^n (z_i - b_i)^2
w2(x_i) over the class of pairs of vectors (a, b) such that a
and b are isotonic and a_i <= b_i for all i = 1, ..., n. We
offer two different approaches to compute the estimates: a
projected subgradient algorithm where the projection is
calculated using a PAVA as well as Dykstra's cyclical
projection algorithm.