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Proposed hybrid filter-wrapper approach.
Since combining of filter and wrapper methods always leads to inaccuracy, application of Partial Mutual Information filter approach is implemented. The Partial Mutual Information filter narrows down traits by removing superfluous or inappropriate traits. Also, use of a wrapper method most preferably the firefly algorithm is used to get the point traits.
Algorithms in practice.
Partial Mutual Information can be determined directly using
(1)
p(y) and p(x) are the marginal probability density functions (pdfs) of X and Y, respectively; and p(x, y) is the joint pdf. However, within a practical context, the true functional forms of the pdfs in (1) are typically unknown. Hence, estimates of the densities are used instead. Substitution of density estimates into a numerical approximation of the integral in (1) gives
(2)
Where f denotes the estimated density based on a sample of n observations of (x, y). Note that the base of the logarithm varies within the literature and use of either 2 or e is often reported, although the natural logarithms assumed in this study unless otherwise stated. Given the form of (2), it follows that efficient and accurate estimation of MI is largely dependent on the technique employed to estimate the marginal and joint pdfs. Non-parametric density estimation techniques are typically considered suitably robust and precise. In particular kernel density estimation (KDE) is used, although it is somewhat computationally intensive compared to alternatives, such as the histogram (Scott, 1992).
f is given by:
(3)
Where denotes the estimate of the pdf at denote sample observations of X: and Kh is some kernel function for which h denotes the kernel bandwidth (or, smoothing parameter). A common choice for Kh is the Gaussian kernel.
(4)
Here, d denotes the number of dimensions of X, Ʃ is the sample covariance matrix, and is the Mahalanobis distance metric, which is given by: (5)
Substituting the expression for the kernel into (3), the estimator for ƒ becomes:
(6)
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