where T is the matrix transpose operator and 0 is a 4 × 4 matrix of zeros.
One can note that C0,0 and C1,1 are symmetric, and t (C1,0) = C0,1 (where t is the matrix transpose operator).
Then, a molecular fingerprint can be formulated as a 256-D vector given by (2) MF = [ A 1 ⋯ A j ⋯ A 256 ] T, where A j (j = 1,2,…, 256) is an integer between 0 and 15, and T is the matrix transpose operator.
RVs, and the superscript T denotes the matrix transpose operation.
is matrix stack operator, is matrix transpose, is Hermitian operation, is complex conjugate, denotes the trace of a matrix, is Frobenius norm, ( is vector norm), denotes the Kronecker product, denotes identity matrix, and defines new symbols.
(2) Calculate the projection matrix, which is given by (8) where " " is the unit matrix and " " is the transpose operator.
where " " is the unit matrix and " " is the transpose operator.
(3) and (5), the copula density can be derived as shown in Eq. (6), where I denotes an identity matrix and Tr denotes transpose operator.
The above equation can be rewritten in matrix notation as: (2) F = arg min F t r ((F - Y ) T H (F - Y ) ) + t r (F T L F ) Here, Y = [ y1, y2,⋯, y n ], F = [ f1, f2,⋯, f n ] ∈ R n × C, H is an n × n diagonal matrix with H ii = 1 if i ≤ l, and H ii = 0 otherwise, L = (I - W ) T (I - W) and I is an n × n identity matrix, tr and T are the matrix trace and transpose operators, respectively.
|A| denotes the determinant of matrix A. The superscripts H, and −1 represent the conjugate transpose operator, and the matrix inverse, respectively.
where I is an M × M identity matrix, H is the Hermitian transpose operator, and φ y b ) is a vector function whose ith element is φ ( y b ) i = ∂ log p ( y i b ) ∂ y i b = ∑ c ∈ S b y i b ∑ b = f c l c | y i b | 2, (16).
