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Gaussian Mixture Model

Module by: Yi-Chieh Wu. E-mail the author

Summary: A description of Gaussian Mixture Models as applied to instrument classification.

Gaussian Mixture Model

A Gaussian Mixture Model (GMM) was used as our classification tool. As our work focused mainly on signal processing, we forgo a rigorous treatment of the mathematics behind the model in favor of a brief description of GMMs and its application to our system.

GMMs belong to the class of pattern recognition systems. They model the probability density function of observed variables using a multivariate Gaussian mixture density. Given a series of inputs, it refines the weights of each distribution through expectation-maximization algorithms.

In this respect, GMMs are very similar to Support Vector Machines and Neural Networks, and all of these models have been used in instrument classification (1). Reported success (2) with GMMs prompted us to use this model for our system.

Recognizing Spectral Patterns

We use 9 features in our recognition program and relied on the GMM to find patterns that would associate these features to the correct instrument. Some of our features consist of a vector (we used 12 MFCC, and tristimulus has 3 components), so we are actually working in 22 dimension space. For convenience, we focus here on recognizing a pattern between the instrument and two of these dimensions, using the first two MFCC coefficients as an example.

Looking at the distribution of features for the three instruments in figure 1, we clearly see that there are some feature differences based on instrument.

Figure 1: Despite the heavy overlap, we see that each instrument dominates different sections of the cepstral space.
Distribution of First Two MFCC Coefficients for Three Instruments
Distribution of First Two MFCC Coefficients for Three Instruments (Graphic1)

GMM detects the patterns in these features and gives us a nice decision rule, as pictured in figure 2. Based on these two features alone, the GMM tells us which instrument most likely played the note, visually represented by the highest peak in the three-dimensional representation.

Figure 2: Gaussian Mixture Model for Clarinet (blue), Saxophone (green), and Trumpet (red). Signals with features falling in a colored area are classified as a particular instrument. (Gray represents indeterminate instrument.)
Two-Parameter Gaussian Mixture Model for Three Instruments
Two-Parameter Gaussian Mixture Model for Three Instruments (Graphic2)

Finally, we note that GMMs have been shown to be useful if features are particularly weak or missing (2). This is of particular importance in polyphonic environments, as harmonics may overlap, thus causing some features to be unreliable measures of the instrument.

References

  1. A. Brenzikofer. “Instrument Recognition and Transcription in Polyphonic Music.” Term Project. Ecole Polytechnique Federale de Lausanne, June 2004. URL: http://www.brenzi.ch/data/murec-report-web.pdf
  2. J. Eggink and G.J. Brown. “A Missing Feature Approach to Instrument Identification in Polyphonic Music,” in IEEE International Conference on Acoustics, Speech, and Signal Processing, Hong Kong, April 2003, 553-556.
  3. D. Ellis. Musical Content Analysis: A Practical Investigation of Singing Detection. URL: http://www.ee.columbia.edu/~dpwe/muscontent/practical/index.html

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