Mixture density networks

Minimization of a sum-of-squares or cross-entropy error function leads to network outputs which approximate the conditional averages of the target data, conditioned on the input vector. For classifications problems, with a suitably chosen target coding scheme, these averages represent the posterior probabilities of class membership, and so can be regarded as optimal. For problems involving the prediction of continuous variables, however, the conditional averages provide only a very limited description of the properties of the target variables. This is particularly true for problems in which the mapping to be learned is multi-valued, as often arises in the solution of inverse problems, since the average of several correct target values is not necessarily itself a correct value. In order to obtain a complete description of the data, for the purposes of predicting the outputs corresponding to new input vectors, we must model the conditional probability distribution of the target data, again conditioned on the input vector. In this paper we introduce a new class of network models obtained by combining a conventional neural network with a mixture density model. The complete system is called a Mixture Density Network, and can in principle represent arbitrary conditional probability distributions in the same way that a conventional neural network can represent arbitrary functions. We demonstrate the effectiveness of Mixture Density Networks using both a toy problem and a problem involving robot inverse kinematics.

Statistical mechanics of low-density parity check error-correcting codes over Galois fields

A variation of low-density parity check (LDPC) error-correcting codes defined over Galois fields (GF(q)) is investigated using statistical physics. A code of this type is characterised by a sparse random parity check matrix composed of C non-zero elements per column. We examine the dependence of the code performance on the value of q, for finite and infinite C values, both in terms of the thermodynamical transition point and the practical decoding phase characterised by the existence of a unique (ferromagnetic) solution. We find different q-dependence in the cases of C = 2 and C ≥ 3; the analytical solutions are in agreement with simulation results, providing a quantitative measure to the improvement in performance obtained using non-binary alphabets.

Prediction of the equilibrium structures and photomagnetic properties of the Prussian blue analogue RbMn[Fe(CN)6] by density functional theory

A periodic density functional theory method using the B3LYP hybrid exchange-correlation potential is applied to the Prussian blue analogue RbMn[Fe(CN)6] to evaluate the suitability of the method for studying, and predicting, the photomagnetic behavior of Prussian blue analogues and related materials. The method allows correct description of the equilibrium structures of the different electronic configurations with regard to the cell parameters and bond distances. In agreement with the experimental data, the calculations have shown that the low-temperature phase (LT; Fe(2+)(t(6)2g, S = 0)-CN-Mn(3+)(t(3)2g e(1)g, S = 2)) is the stable phase at low temperature instead of the high-temperature phase (HT; Fe(3+)(t(5)2g, S = 1/2)-CN-Mn(2+)(t(3)2g e(2)g, S = 5/2)). Additionally, the method gives an estimation for the enthalpy difference (HT LT) with a value of 143 J mol(-1) K(-1). The comparison of our calculations with experimental data from the literature and from our calorimetric and X-ray photoelectron spectroscopy measurements on the Rb0.97Mn[Fe(CN)6]0.98 x 1.03 H2O compound is analyzed, and in general, a satisfactory agreement is obtained. The method also predicts the metastable nature of the electronic configuration of the high-temperature phase, a necessary condition to photoinduce that phase at low temperatures. It gives a photoactivation energy of 2.36 eV, which is in agreement with photoinduced demagnetization produced by a green laser.

An assessment of the performance of high density polyethylene copolymers for natural gas distribution pipes

The total thermoplastics pipe market in west Europe is estimated at 900,000 metric tonnes for 1977 and is projected to grow to some 1.3 million tonnes of predominantly PVC and polyolefins pipe by 1985. By that time, polyethylene for gas distribution pipe and fittings will represent some 30% of the total polyethylene pipe market. The performance characteristics of a high density polyethylene are significantly influenced by both molecular weight and type of comonomer; the major influences being in the long-term hoop stress resistance and the environmental stress cracking resistance. Minor amounts of hexene-1 are more effective than comonomers lower in the homologous series, although there is some sacrifice of density related properties. A synergistic improvement is obtained by combining molecular weight increase with copolymerisation. The Long-term design strength of polyethylene copolymers can be determined from hoop stress measurement at elevated temperatures and by means of a separation factor of approximate value 22, extrapolation can be made to room temperature performance for a water environment. A polyethylene of black composition has a sufficiently improved performance over yellow pigmented pipe to cast doubts on the validity of internationally specifying yellow coded pipe for gas distribution service. The chemical environment (condensate formation) that can exist in natural gas distribution networks has a deleterious effect on the pipe performance the reduction amounting to at least two decades in log time. Desorption of such condensate is very slow and the influence of the more aggressive aromatic components is to lead to premature stress cracking. For natural gas distribution purposes, the design stress rating should be 39 Kg/cm2 for polyethylenes in the molecular weight range of 150 - 200,000 and 55 Kg/cm2 for higher molecular weight materials.