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## Problem setting

The working hypothesis to be tested was that the NDFA algorithm should be able to find a representation for the observations which facilitates the prediction of the observations, i.e., a representation where modelling the dynamics of the underlying data generating process is easier than directly for the original observations.

Data was generated by first defining the underlying dynamic process and then mapping some of the states of the process onto observations by one of the random MLP networks which were used in [4].

The underlying dynamic process was constructed by combining three independent dynamic processes. One of the processes was harmonic oscillator with angular velocity 1/3. The harmonic oscillator has a two-dimensional state representation and linear dynamics. Notice, however, that if the harmonic oscillator is modelled with linear dynamics, the gain has to be exactly one. Otherwise the oscillations will either decay or grow exponentially for gains less than or greater than one, respectively. A robust model for a harmonic oscillator with a constant amplitude therefore needs to be nonlinear.

The two other processes were chosen to be Lorenz processes [5]. The Lorenz system has a three-dimensional state space whose dynamics is governed by the following set of differential equation:

 = (16) = (17) = (18)

The parameter vectors for the processes were chosen to be and . The time evolution of the state for the first Lorenz process is shown in figure 3.

The ability to find the underlying factors which have generated the observations was already demonstrated in [4]. To make the problem more difficult, one dimension of each of the three elementary processes was hidden. As the original process has a total of eight states, five states are left from which the data was generated nonlinearly. In [4], some of the experiments used a random MLP network with inverse hyperbolic sine activation functions to generate ten-dimensional observations from five inputs. The same MLP network was used here. Like in [4], the data set consisted of 1000 observation.

Figure 4 shows the eight states of the underlying dynamical system, the five projections made from them and the noisy observations obtained from the projections by the random MLP network. The standard deviation of the observation noise was 0.1 while the standard deviation of the signal was normalised to unity.

Next: Finding the underlying process Up: Results Previous: Results
Harri Valpola
2000-10-17