Probabilistic Simplex Component Analysis by Importance Sampling

In this paper we consider the problem of linear unmixing hidden random variables defined over the simplex with additive Gaussian noise, also known as probabilistic simplex component analysis (PRISM). Previous solutions to tackle this challenging problem were based on geometrical approaches or computationally intensive variational methods. In contrast, we propose a conventional expectation maximization (EM) algorithm which embeds importance sampling. For this purpose, the proposal distribution is chosen as a simple surrogate distribution of the target posterior that is guaranteed to lie in the simplex. This distribution is based on the Gaussian linear minimum mean squared error (LMMSE) approximation which is accurate at high signal-to-noise ratio. Numerical experiments in different settings demonstrate the advantages of this adaptive surrogate over state-of-the-art methods

(E)-1-(5-Hy­droxy-2,2-dimethyl-2H-chromen-6-yl)-3-(3,4,5-trimeth­oxy­phen­yl)prop-2-en-1-one

The title compound, C23H24O6, crystallizes with two independent mol­ecules (A and B) in the asymmetric unit. The dihedral angles between the benzopyran ring and the α,β-unsaturated ketone unit and between the α,β-unsaturated ketone group and the benzene ring are 9.4 (10) and 12.96 (13)°, respectively, in mol­ecule A and 1.40 (17) and 4.44 (17)°, respectively, in mol­ecule B. The two meth­oxy groups at the meta positions of the benzene ring are close to being coplanar with the ring [C—O—C—C = 6.2 (3) and −1.4 (3)° in mol­ecule A and −4.2 (4) and 3.7 (3)° in mol­ecule B], whereas the third meth­oxy group, at the para position, is (+)-anti­clinal with respect to the benzene ring [C—O—C—C = 81.7 (3)°] in mol­ecule A and is (−)-synclinal with respect to the benzene ring [C—O—C—C = −103.2 (3)°] in mol­ecule B. In both independent mol­ecules, the hy­droxy group is involved in an intra­molecular O—H⋯O hydrogen bond

3,4-Dihydroxy­phenyl 3,4,5-trimethoxy­benzoate

In the title compound, C16H16O7, the dihedral angle between the two benzene rings is 82.02 (7)°. The crystal structure is stabilized by inter­molecular O—H⋯O hydrogen bonds, which link the mol­ecules into a two-dimensional network

(E)-1-(4-Amino­phen­yl)-3-(naphthalen-2-yl)prop-2-en-1-one

The mol­ecule of the title chalcone derivative, C19H15NO, exists in a trans configuration with respect to the C=C double bond. The mol­ecule is slightly twisted with a dihedral angle of 6.12 (12)° between the benzene ring and the naphthalene ring system. The prop-2-en-1-one bridge is nearly planar, with an r.m.s. deviation of 0.0194 (2), and makes dihedral angles of 8.05 (19) and 11.47 (18)° with the benzene ring and the naphthalene ring system, respectively. In the crystal, mol­ecules are linked by N—H⋯O hydrogen bonds into chains along the b axis. Weak N—H⋯π and C—H⋯π inter­actions and a short N⋯O contact [2.974 (4) Å] are also observed

4-Hydroxy­phenyl 4-fluoro­benzoate

In the title compound, C13H9FO3, the dihedral angle between the two benzene rings is 59.86 (4)°. In the crystal, inter­molecular O—H⋯H hydrogen bonds lead to molecular chains propagating in [010]

N-(3,4-Difluoro­phen­yl)-N′-(2,5-di­methoxy­phen­yl)urea

In the title compound, C15H14F2N2O3, the dihedral angle between the benzene rings is 64.5 (1)°. One F atom is disordered over two meta positions, with occupancy factors of 0.72 and 0.28. In the crystal, mol­ecules are linked by N—H⋯O hydrogen bonds involving two N—H and one C=O groups of the urea central fragment, leading to a supra­molecular chain along [011]

N-(3,4-Difluoro­phen­yl)-2-(3,4-dimethoxy­phen­yl)acetamide

In the title amide, C16H15F2NO3, the dihedral angle between the benzene rings is 53.7 (1)°. Mol­ecules are linked in the crystal structure by an inter­molecular N—H⋯O hydrogen bond involving N—H and C=O functionalities of the amide group. A one-dimensional network is thus formed along the [001] direction. No significant inter­chain contacts are observed

(E)-1-(Furan-2-yl)-3-(2,4,5-trimeth­oxy­phen­yl)prop-2-en-1-one

In the title chalcone derivative, C16H16O5, the dihedral angle between the furan and benzene rings is 2.06 (17)°. The two meth­oxy groups at the ortho and para positions are essentially coplanar with the benzene ring [C—O—C—C angles = −1.0 (5) and 178.5 (3)°], whereas the third one at the meta position is slightly twisted [C—O—C—C = 9.6 (5)°]. In the crystal, weak C—H⋯O inter­actions link the mol­ecules into a sheet parallel to (02). An inter­molecular π–π inter­action between the furan and benzene rings is present [centroid–centroid distance = 3.772 (2) Å]. A short C⋯C contact [3.173 (5) Å] is also observed between neighbouring furan rings

(E)-3-(2,6-Dichloro­phen­yl)-1-(4-methoxy­phen­yl)prop-2-en-1-one

In the title compound, C16H12Cl2O2, the dichloro­phenyl and methoxy­phenyl groups are linked by a prop-2-en-1-one group. The C=C double bond is trans configured. The mol­ecule is not planar, as can be seen from the dihedral angle of 6.21 (7)° between the planes of the two rings. The crystal structure can be described by two types of crossed layers which are parallel to (110) and (10)