La resilencia mide la vulnerabilidad del sistema a los cambios o choques Cuanto

The initial interpretation of the findings in SPSS non-parametric testing techniques that were achieved, indicated a need for further analysis to maximise the results. This would be done by selecting the most appropriate multivariate technique(s) available. The researcher sought advice from the high performance computing unit at the Queensland University of Technology, Brisbane on which analytical path to take. The advice was to the effect that the most appropriate technique/selection is by adherence to a set of common rules. These are to:

1. Acknowledge the assumptions of the technique(s); 2. Know the data characteristics and requirements; and 3. Determine the final use of the outcome information.

Evidence of satisfying but not maximising the requirements of the study in the SPSS output data showed all the signs of meeting the above criteria, but not exceeding it, so the researcher pursued this further. The need for ‘proving’ replication in the survey through triangulation and in achieving analytical generalisation was where the research output in SPSS appeared wanting. This was why decision tree analysis in Classification and Regression Trees (CART) models by Breiman et al., (1984) and Multivariate Adaptive Regression Splines (MARS) models by Friedman (1991) would be expected to meet these requirements in this study.

4.2.1 CART Models in Non-Parametric Statistics

Consequently, to gain highly relevant insights from the survey sufficient to inform the research, CART models were constructed using multinomial categorical predictors, that are important in finding associations and relationships in the data (Breiman, et al., 1984). Set variables in CART models have an added bonus (by comparison with conventional regression analyses), in that they uncover hidden meaning in the data through its data mining application in the tree structures which proved very fruitful for this research. As CART was used for classification purposes and not regression (due to the research design), MARS models in a non-parametric local regression equation were used in order to add to the efficacy and stability of the variables under review and which would assist in achieving triangulation of the CART data sets (Friedman, 1991; Steinberg and Colla, 1995).

With the conventions of hypothesis testing set aside in substitution for research propositions, the use of CART and MARS models in a non-linear, non-parametric application to data analysis increased the robustness of the findings in this study. The technical aspects of CART and MARS models are very sound and, thus are regarded as an appropriate analytical technique in analysis (Breiman et al., 1984; Friedman, 1991).

4.2.1 MARS Models in Non-Parametric Statistics

Multiple Adaptive Regression Splines (MARS) is a method for flexible modelling of high dimensional data. In brief overview here, the MARS model takes the form of an expansion in product spline Basis functions, where the number of

Basis functions as well as the parameters associated with each one (product degree

and knot locations) are automatically determined by the data. This procedure is motivated by the recursive partitioning approach to regression, and shares its attractive properties. Unlike recursive partitioning, however, this method provides more power and flexibility to model relationships that are nearly additive or involve interactions in, at most, a few variables. In addition, the model can be represented in a form that separately identifies the additive contributions and those associated with the different multivariate interaction (Besley, Kuh and Welsch, 1980; Friedman, 1991).

The goal is to model the dependence of a response variable y on one or more predictor variables given realisations (data) { } . The system that generated the data is presumed to be described by:

( )

over the domain of the ( ) n containing the data. The single valued deterministic function of , of its -dimensional argument, captures the joint predictive relationship of on . The additive stochastic component of whose expected value is defined to be zero usually reflects the dependence of on quantities other than that are neither controlled nor observed. The aim of non-parametric regression analysis is to use the data to construct a function ( ) that can serve as reasonable approximation to ( ) over the domain of interest. The notion of reasonableness depends on the purpose for which the approximation is to be used. In nearly all applications, however, accuracy is important. Lack of accuracy is often defined by an integral error or expected error, both with their own formulas (Scott, 1992).

Recursive Partitioning Regression

With regard to recursive partitioning regression in this (non-parametric design) study, these are disjoint sub regions representing a partition of . The goal is to use the data to simultaneously estimate a good set of sub-regions and the parameters associated with the separate functions in each sub-region. Partitioning is accomplished through the recursive splitting of previous sub-regions. The starting region is the entire domain . At each stage of the partitioning, all existing sub- regions are each optimally split into two sibling sub-regions. The split is jointly optimised using a goodness-of-fit criterion on the resulting approximation. The recursive subdivision is continued until the large number of subregions are generated, as in the structured trees created (Friedman, 1991).

Recursive partitioning based on linear functions, basically lacks a local variable subset selection feature. A global variable subset selection emerges as a natural consequence. This tended to limit its power (and interpretability) and was probably the main reason contributing to its lack of popularity (Besley, Kuh and Welsch, 1980). To overcome this problem, recursive partitioning is a powerful tool when a piecewise constant approximation is used (Breiman et al., 1984), as was applied in this study. It has the ability to exploit lower local variables in the tree even though they may be dependent upon a large number of variables (globally) higher in the tree, that is, even though the function of (1) may strongly depend on only a few of them. This ability comes from the splitting rules that become more and more local, the lower they are in the tree. Variables that have less influence on the response are less likely to be used for splitting. This gives rise to the local variable subset selection.

Another property that recursive partitioning regression exploits, is the marginal consequences of interaction effects (Steinberg and Colla, 1995). That is, a local intrinsic dependence on several variables, when best approximated by an additive function, does not lead to a constant model. This is nearly always the case. Recursive partitioning models using piecewise constant approximations that are fairly interpretable owing to the fact that they are very simple and can be represented by a binary tree as was developed in this study. Although recursive partitioning is the

most adaptive of the methods for multivariate function approximation, it has some limitations. These are briefly:

① the approximating function is discontinuous at the sub-region boundaries; and

② certain types of linear functions with more than a few non-zero coefficients are difficult to approximate (with a piecewise approximation and additive functions in more than a few variables), as piecewise or constant or piecewise linear approximation.

Recursive partitioning regression is generally viewed as a geometrical procedure. Its framework provides the best insight into the properties mentioned earlier and can be viewed in a more conventional light as a stepwise regression procedure. The aim is to produce an equivalent model to the general format by replacing the geometrical concepts of regions and splitting with the arithmetic notions of adding and multiplying (Friedman, 1991).

Multivariate Adaptive Regression Splines

This area describes the multivariate adaptive regression splines (MARS) approach to a multivariate non-parametric regression in this study. The goal of this procedure was to take advantage of its ability to compute in high dimensional settings where adaptive computation is used and advancing the earlier tests in SPSS non-parametric. It is most easily understood through its connections with recursive partitioning regression and has been developed consequently as a series of generalisations to that procedure.

In this chapter are the MARS models that commence with a series of Basis functions. The starting point is to cast the approximation in the form of an expansion from those set of Basis functions. The recursive partitioning regression model takes the form as identified by the following formula:

(1) if x , then ( ) gm

(

).

Here the {Rm} are disjoint subregions representing a partition of D. The

functions gm are generally taken to be of quite simple parametric form. The most common is a constant function:

(2) gm

(

)

Adaptive regression splines are an approximation from the recursive partioning regression model:

(3) ( ) ∑ Bm (x) The Basis functions B take the form:

(4) B ( ) [ ]

Where I is an indicator function having the value one if the argument is true, and, otherwise zero, if it is false. The { } are the coefficients of the expansion whose values are jointly adjusted to give the best fit to the data. The { } are the same sub-regions of the covariate space as in (1), (2). Since these regions are disjoint, only one Basis function is non zero for any point so that equations (3), (4) is equivalent to (1), (2). The aim of recursive partitioning is not only to adjust the coefficient values to best fit the data, but also to derive a good set of Basis functions (sub- regions) based on the data at hand.

For more details on CART and MARS analytical techniques and procedures, see Appendix F.