Speech Spectra

As a real world data set we used speech spectra. The data set consisted of 11200 21 dimensional samples which corresponds to 90 seconds of continuous human speech. The first 10000 samples were used to train a seven dimensional state-space model with the method from [1] and the rest of the data was used in the experiments. This data set poses a somewhat different problem from the double inverted pendulum system. The nonlinearities are not as strong as in the first experiment but the dimensionality of the observation and state spaces are higher, which emphasises the scalability of the methods.

The test data set was divided into three parts each consisting of 300 samples and all the algorithms were run for each data set with four random initialisations. The final results represent an average over both the different data sets and initialisations.

Since the true state is unknown in this experiment, the mean square error of the reconstruction of missing data was used to compare the different algorithms. Experiments were done with sets of both 3 and 30 consecutive missing samples. The ability to cope with missing values is very important when only partial observations are available or in the case of failures in the observation process. It also has interesting applications in the field of control as reported in [12].

Results can be seen in Figure 2. When missing values are present, especially in the case of the large gap size, the proposed algorithm performs clearly better than the rest of the compared algorithms. Compared to the double inverted pendulum data set, the stability issues with IEKS and IUKS were not as severe, but neither method could cope very well with long gaps of missing values.

Figure 2: Inference with the speech data and missing values. On the top one of the data sets used in the experiments (missing values marked in black), on the bottom root mean square error plotted against computation time. Left side figures use a small gap size, right side figures a large gap size.