SummaryA maximum likelihood estimation procedure of Hawkes' self-exciting point process model is proposed with explicit presentations of the loglikelihood of the model and its gradient and Hessian. A simulation method of the process is also presented. Some numerical results are given.

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Dimethyl ether (DME) from natural gas or coal via syngas (3CO + 3H 2 f DME + CO 2 ) attracts much attention as a high quality diesel fuel of the next generation. Considering the compact process based on the small-scale carbon resources, both low-pressure operation (at 1-3 MPa) and high one-pass conversion (90% CO conversion) are required to produce DME economically. A temperature gradient reactor (TGR) was effective for overcoming both the equilibrium limit of the reaction at high temperature and the low activity of the catalyst at low temperature. For example, 90% CO conversion and high STY (1.1 kg-MeOH eq./kg-cat./h) was attained at the same time in TGR at 280-240 °C, 3 MPa. For higher performance of the TGR, in the next step, optimization of the temperature gradient was required. A homemade program according to a genetic algorithm (GA) was used for the optimization. The catalyst bed was divided into 5 zones in series. The temperature of each zone was encoded to "gene" and the fitness of the "gene" was evaluated by CO conversion obtained in the reactor of which temperatures were set according to the gene. After a few generations of "evolution", CO conversion (70%) higher than that in a conventional isothermal reactor (66%) was achieved at 1 MPa. For higher CO conversion, neural network, trained by the results in the preceding generations, was used to evaluate the "gene". By this combination of GA and neural network, evolution of the temperature profile was accelerated and successfully optimized to give 71% CO conversion.

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