From: Bi-linear matrix-variate analyses, integrative hypothesis tests, and case-control studies
IHT types | Applications |
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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. |