Optimality

Regression trees 215
of the tree. The piecewise best simple linear GUIDE model can be expressed as
(3.4)
ˆy = (14.14246 + 0.4875417x4)(1−x2)(1−x3)/4
+ 14.0075(1−x2)(1 + x3)/4
+ (14.24752 + 0.2299792x4)(1 + x2)/2
= 14.16125 + 0.23688x4 + 0.12189x3x4(x2− 1)
+ 0.08627x2 + 0.03374x3(x2− 1)−0.00690x2x4.
Figure 3, which superimposes the fitted functions from the three leaf nodes, offers
a more vivid way to understand the interactions. It shows that changing the level of
D from−to + never decreases the predicted mean yield and that the latter varies
less if D =− than if D = +. The same tree model is obtained if we fit a piecewise
multiple linear GUIDE model using forward and backward stepwise regression to
select variables in each node.
A simulation experiment was carried out to compare the PMSE of the methods.
Four models were employed, as shown in Table 2. Instead of performing the simula-
Y
13.8 14.0 14.2 14.4 14.6
B = C =
B = +
+
0.0 0.5 1.0
Fig 3. Fitted values versus x4 (D) for the piecewise simple linear GUIDE model shown on the
right side of Figure 2.
Table 2
Simulation models for a 2 4 design; the βi’s are uniformly distributed and ε is normally
distributed with mean 0 and variance 0.25; U(a, b) denotes a uniform distribution on the
interval (a, b); ε and the βi’s are mutually independent
Name Simulation model β distribution
Null y = ε
Unif y = β1x1 + β2x2 + β3x3 + β4x4 + β5x1x2 + β6x1x3 +
β7x1x4+β8x2x3+β9x2x4+β10x3x4+β11x1x2x3+β12x1x2x4+
β13x1x3x4 + β14x2x3x4 + β15x1x2x3x4 + ε
U(−1/4, 1/4)
Exp y = exp(β1x1 + β2x2 + β3x3 + β4x4 + ε) U(−1, 1)
Hier y = β1x1 + β2x2 + β3x3 + β4x4 + β1β2x1x2 + β1β3x1x3 +
β1β4x1x4+β2β3x2x3+β2β4x2x4+β3β4x3x4+β1β2β3x1x2x3+
U(−1,1)
β1β2β4x1x2x4 + β1β3β4x1x3x4 + β2β3β4x2x3x4 +
β1β2β3β4x1x2x3x4 + ε
X 4