5. Análisis de resultados
5.2. Categoría Competencia argumentativa
5.2.2 Subcategoría – Interacciones en el aula y participación de los estudiantes
1: Initialize M=0d×d and H(i)=0n1×mfor i=1, . . . ,d.
2: %calculate and pre-store the univariate KDE 3: for k=1, . . . ,d do 4: for k′=1, . . . ,m do 5: bp(x(kk′))← 1 n1s∈D1
∑
1 h1 K X (s) k −x (k′) k h1 ! 6: for k′=1, . . . ,m do7: %calculate the components used for the bivariate KDE 8: for i′=1, . . . ,n1do 9: for i=1, . . . ,d do 10: H(i)(i′,k′)← 1 h2 K X i′ i −x (k′) i h2 !
11: %calculate the mutual information matrix 12: forℓ′=1, . . . ,m do 13: for i=1, . . . ,d−1 do 14: for j=i+1, . . . ,d do 15: bp(xi(k′),x(jℓ′))←0 16: for i′=1, . . . ,n1do 17: pb(x(ik′),x(jℓ′))←pb(x(ik′),x(jℓ′)) +H(i)(i′,k′)·H(j)(i′, ℓ′) 18: pb(xi(k′),x(jℓ′))←bp(x(ik′),x(jℓ′))/n1 19: M(i,j)←M(i,j) + 1 m2pb(x (k′) i ,x (ℓ′) j )·log b p(xi(k′),x(jℓ′))/(pb(x(ik′))·pb(x(jℓ′)))
evaluate the kernel density estimates on
D
1. The space complexity for the Chow-Liu algorithm isO(d2), and thus the total space complexity is O(d2+m2n1).
The quadratic time and space complexity in the number of variables d is acceptable for many practical applications but can be prohibitive when the dimension d is large. The main bottleneck is to calculate the empirical mutual information matrix M. Due to the use of the kernel density estimate, the time complexity is O(d2m2n1). The straightforward implementation in Algorithm 1 is conceptually easy but computationally inefficient, due to many redundant operations. For example, in the nested for loop, many components of the bivariate and univariate kernel density estimates are repeatedly evaluated. In Algorithm 6, we suggest an alternative method which can significantly reduce such redundancy at the price of increased but still affordable space complexity.
The main technique used in Algorithm 6 is to change the order of the multiple nested for loops, combined with some pre-calculation. This algorithm can significantly boost the empirical perfor- mance, although the worst case time complexity remains the same. An alternative suggested by Bach and Jordan (2003) is to approximate the mutual information, although this would require fur- ther analysis and justification.
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