Analytic Long Term Forecasting with Periodic Gaussian Processes

Abstract: In many application domains such as weather forecasting, robotics and machine learning we need to model, predict and analyze the evolution of periodic systems. For instance, time series applications that follow periodic patterns appear in climatology where the CO2 emissions and temperature changes follow periodic or quasi-periodic patterns. Another example can be in robotics where the joint angle of a rotating robotic arm follows a periodic pattern. It is often very important to make long term prediction of the evolution of such systems. For modeling and prediction purposes, Gaussian processes are powerful methods, which can be adjusted based on the properties of the problem at hand. Gaussian processes belong to the class of probabilistic kernel methods, where the kernels encode the characteristics of the problems into the models. In case of the systems with periodic evolution, taking the periodicity into account can simplifies the problem considerably. The Gaussian process models can account for the periodicity by using a periodic kernel. Long term predictions need to deal with uncertain points, which can be expressed by a distribution rather than a deterministic point. Unlike the deterministic points, prediction at uncertain points is analytically intractable for the Gaussian processes. However, there are approximation methods that allow for dealing with uncertainty in an analytic closed form, such as moment matching. However, only some particular kernels allow for analytic moment matching. The standard periodic kernel does not allow for analytic moment matching when performing long term predictions. This work presents an analytic approximation method for long term forecasting in periodic systems. We present a different parametrization of the standard periodic kernel, which allows us to approximate moment matching in an analytic closed form. We evaluate our approximate method on different periodic systems. The results indicate that the proposed method is valuable for the long term forecasting of periodic processes.