## 5 Related Work  
### GPs in Sound Synthesis and Audio Analysis  
GPs have been used to synthesize natural sounds, to reproduce acoustic instrument sound, to synthesize speech and to analyze audio and music signals. [PERSON] et al. [17] used Latent Force Models to learn modal synthesis parameters. Modal synthesis is a physically inspired synthesis method [18] (ch. 4). Latent Force Models [19] combine GPs with physical models. The latent state is sampled from a GP. The observations depend on the latent state via a mechanistic model that is expressed through differential equations. An example is forces acting on springs: the forces over time are sampled from a GP, the observations are the spring positions over time, and the physical model is a second-order differential equation. Latent Force Models can be applied in various areas, including biology, motion capture and geostatistics. In the model by [PERSON] et al. [17], the system learns the stiffness and the damping constants of a mass-spring system for each mode. The modes are peak-picked from a spectrogram. The mass-spring system is modeled with a differential equation. It relates an unknown excitation function, which is sampled from a GP, to the measured amplitude over time of the mode and the mode-specific stiffness and damping constants. [PERSON] et al. used this to model and synthesize clarinet and oboe sounds (sound examples: [http://c4 dm.eecs.qmul.ac.uk/audioengineering/latent-force-synthesis](http://c4 dm.eecs.qmul.ac.uk/audioengineering/latent-force-synthesis)). It is possible to interpolate the parameters and morph between two sounds. Further interaction possibilities are provided: the excitation curves can be hand-drawn [17], or the user can control the mean function of the posterior with real-time MIDI messages [20]. A further Latent Force Model by [PERSON] et al. [21] uses a difference equation that models temporal correlation by incorporating values from previous time steps. The temporal contribution is modulated by feedback coefficients (for the influence of previous output) and lag coefficients (for the influence of previous input). In combination with damping coefficients for exponential decay, amplitude envelopes are predicted for multiple frequency bands. The system was trained to re-synthesize natural sounds. GPs have also been used to synthesize speech. For example, [PERSON] and [PERSON] [22] used GP regression to predict frame-wise the acoustic features for parametric speech synthesis. The similarity of linguistic and position features is modeled in the covariance matrix. The model predicts duration, fundamental frequency, mel-cepstrum, band aperiodicity, and whether the sound is voiced or unvoiced. GPs have been used for music information retrieval. [PERSON] and [PERSON] used GPs to perform pitch estimation and to impute short gaps of missing audio data [23]. They modeled the musical signal by an additive superposition of periodic kernels, which are localized in time with \"change windows\". For these change windows, they made use of the fact that for a given kernel k and any function \(\phi\), the expression \(\phi(x_{1})k(x_{1},x_{2})\phi(x_{2})\) is again a kernel. In another study, [PERSON] et al. used GPs for source separation in the time-domain. [24]. They modeled the different spectral characteristics of signals with spectral mixture kernels. It is also possible to extract high-level information from audio signals with GPs. [PERSON] and [PERSON] [25] applied GP regression to music emotion recognition to predict valence and arousal from standard audio features such as MFCC. With the same features, GP classification predicts the genre of the music. Analytic and synthetic approaches can be combined elegantly in the Bayesian framework. [PERSON] [26] used GPs to develop Bayesian methods for statistical audio analysis. For this purpose, generative probabilistic models of sounds are formulated. Bayesian inference is then used on the generative models to analyze natural sounds and auditory scenes.