This paper presents an improved nonlinear mixture density approach to modeling the time-dependent variance in time series. First, we elaborate a recurrent mixture density network for explicit modeling of the time conditional mixing coefficients, as well as the means and variances of its Gaussian mixture components. Second, we derive training equations with which all the network weights are inferred in the maximum likelihood framework. Crucially, we calculate temporal derivatives through time for dynamic estimation of the variance network parameters. Experimental results show that, when compared with a traditional linear heteroskedastic model, as well as with the nonlinear mixture density network trained with static derivatives, our dynamic recurrent network converges to more accurate results with better statistical characteristics and economic performance.
Nikolaev, N. I., Tino, P., & Smirnov, E. N. (2013). Time-dependent series variance learning with recurrent mixture density networks. Neurocomputing, 122, 501-512. https://doi.org/10.1016/j.neucom.2013.05.014