Tree-structured smooth transition regression models

Abstract

ABSTRACT. The goal of this paper is to introduce a tree-based model that combines aspects of CART (Classi- ses. An alternative specification strategy based on a 10-fold cross-validation is also discussed and a detailed Monte Carlo experiment is carried out to evaluate the performance of the proposed methodology in comparison with standard techniques. The STR-Tree model outperforms CART when the correct selection of the architecture of simulated trees is considered. Furthermore, the LM test seems to be a promising alternative to 10-fold cross-validation. When put into proof with real datasets, the STR-Tree model has a superior predictive ability than CART.

Identification of Changes in Mean with Regression Trees: An Application to Market Research

“…It is well-known that tree-based procedures over-fit small data sets (Cooper, 1998;da Rosa et al, 2008). This can be seen in several figures, for example the left panel of Figure 4.…”

Approximations by Smooth Transitions in Binary Space Partitions

“…The STR-Tree model, proposed by Rosa et al in [5], takes advantage of much of the CART structure presented above, but substitutes the sharp splits of the CART model by smooth splits. In the STR-Tree regression model the indicator function I in equation 1 is substituted by a logistic function, parameterized by λ > 0, b ∈ R and j = {1, ...n}, defined as: where the e j is the j th canonical vector.…”

“…In [5] the authors propose to use a least squares approach in order to determine the better c i that minimize the sum of the squared residuals for all input points, once the rectangular partition of the space is given.…”

“…This new tree model, called the Smooth Transition Regression Tree (STRTree), substitutes the indicator function on the CART regression model by a smooth fuzzy membership function. It has also been applied to time series analysis with multipleregimes [5]. However, similarly to the CART model, it restricts the hyperplanes to be axis-aligned, reducing the flexibility of the approximation.…”

“…To solve this problem, Rosa et al proposed in [5] other tree regression model that combines the CART model with the Smooth Transition Regression (STR), which is used in non-linear time series analysis [9]. This new tree model, called the Smooth Transition Regression Tree (STRTree), substitutes the indicator function on the CART regression model by a smooth fuzzy membership function.…”