Table 8 Several IHT Applications

Model based and Mix-modelled

(a) Starting at the case that X _t is degenerated into an 1×2 matrix, we conduct the Hotelling test by Equation (2) and its extension K L _sum in Equation (31), in comparison with both univariate t-test and a paired t-test. (b) For the general case with k≥2, we conduct a matrix-variate test by Equation (28), as well as by the matrix-variate counterparts of K L _1,0, K L _sum , and K L _{s u m∗}, in comparison with not only the Hotelling’s T-square test on the k dimensional vector x _t obtained from Equation (100) but also the paired Hotelling’s T-square test on 2×k matrix-variate samples of X _t . (c) Considering each sample X _t in a 2×k matrix, we investigate the bi-linear discriminant analysis by Equations 18, 33, and 34, in comparison with the classic FDA by Equation (11) on the k dimensional vector x _t obtained from Equation (100). (d) Investigate the generalised bi-linear discriminant analysis by Equations 40, 41, and 34. For simplicity, we get v _i ,i=1,⋯,d by Equation (43) and then solve w by Equation (34). When k becomes too big, we further regularise the learning of v _i by minimising \( J_y=\frac {\alpha _{0} \sigma _{0}^{y\ 2} +\alpha _{1}\sigma _{1}^{y\ 2}} {(c^{y}_{0} -c^{y}_{1})^{2}}+ \sum _{i=1}^{m} \gamma _{i} \sum _{j=1}^{d} |u_{i}^{(j)} |^{q}, \) with q=2 for Tikhonov, q=1 for sparse learning.

Boundary based and Mix-modelled

(a) Consider a logistic regression by Equation (3) with w in one of the ways given in Table 4, we test Equation (5) by the Rao’s score Equation (8), and get ε _C by Equation (44), and ε _B by the p-value with one of choices in Table 2. (b) Extend all the above studies on Equation (3) with y _t =w ^T x _t replaced by the bi-linear form Equation (18). (c) Make a survival analysis via the Cox regression by Equation (13) in comparison with its bi-linear extension by Equations (18) or (40). Again, IHT is made by ε _D , ε _C , and ε _B in a way similar to the above.

BYY harmony

(a) Use either Algorithm 9 in Ref. (Xu, 2015) to get α ⁽ⁱ⁾,c ⁽ⁱ⁾, Σ ⁽ⁱ⁾,i=0,1 or Algorithm ?? to get α ⁽ⁱ⁾,C ⁽ⁱ⁾,Σ ⁽ⁱ⁾,Ω ⁽ⁱ⁾,i=0,1 for model based IHT. (b) Perform the procedure given in Table 5 for training, testing and validating in a small size of samples.