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Statistical Methods for Scene and Event Classification

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Computational Analysis of Sound Scenes and Events

Abstract

This chapter surveys methods for pattern classification in audio data. Broadly speaking, these methods take as input some representation of audio, typically the raw waveform or a time-frequency spectrogram, and produce semantically meaningful classification of its contents. We begin with a brief overview of statistical modeling, supervised machine learning, and model validation. This is followed by a survey of discriminative models for binary and multi-class classification problems. Next, we provide an overview of generative probabilistic models, including both maximum likelihood and Bayesian parameter estimation. We focus specifically on Gaussian mixture models and hidden Markov models, and their application to audio and time-series data. We then describe modern deep learning architectures, including convolutional networks, different variants of recurrent neural networks, and hybrid models. Finally, we survey model-agnostic techniques for improving the stability of classifiers.

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Notes

  1. 1.

    The notation \( \mathbf{P}_{\mathcal{D}} \) denotes the probability mass (or density) with respect to distribution \( \mathcal{D} \), and \( \mathbf{E}_{\mathcal{D}} \) denotes the expectation with respect to distribution \( \mathcal{D} \).

  2. 2.

    Quantifying the relationships between (5.5), (5.4), and (5.3) lies within the purview of statistics and computational learning theory, and is beyond the scope of this text. We refer interested readers to [48, 106] for an introduction to the subject.

  3. 3.

    SVM scores can be converted into probabilities via Platt scaling [94] or isotonic regression [122], but these methods require additional modeling and calibration.

  4. 4.

    The notion of independence for multi-label problems will be treated more thoroughly when we develop deep learning models.

  5. 5.

    The factor of 1∕n is not strictly necessary here, but are included for consistency with (5.6).

  6. 6.

    \( \mathbb{S}_{++}^{d} \) denotes the set of d × d positive definite matrices: Hermitian matrices with strictly positive eigenvalues.

  7. 7.

    A probability distribution P[θ] is a conjugate prior if the posterior P[θ | S] has the same form as the prior P[θ] [97].

  8. 8.

    Note that although we use T to denote the length of an arbitrary sequence x, it is not required that all sequences have the same length.

  9. 9.

    For ease of notation, we denote the initial state distribution as \( \mathbf{P}_{\theta }\left [z[1]\,\middle\vert \,z[0]\right ] \), rather than the unconditional form \( \mathbf{P}_{\theta }\left [z[1]\right ] \).

  10. 10.

    The well-known Baum–Welch algorithm for HMM parameter estimation is a special case of expectation-maximization [96].

  11. 11.

    Some authors refer to the layer dimension d i as width. This terminology can be confusing when applied to spatio-temporal data as in Sect. 5.4.3, so we will use dimension to indicate d i and retain width to describe a spatial or temporal extent of data.

  12. 12.

    To see this, observe that if ρ i is omitted, then the full model f(x | θ) is a composition of affine functions, which is itself an affine function, albeit one with rank constraints imposed by the sequence of layer dimensions.

  13. 13.

    Note that batch normalization accomplishes this scaling implicitly by estimating these statistics during training [68].

  14. 14.

    A valid-mode convolution is one in which the response is computed only at positions where the signal z and filter w fully overlap. For \( z \in \mathbb{R}^{T} \) and \( w \in \mathbb{R}^{n} \), the valid convolution \( w {\ast} z \in \mathbb{R}^{T-n+1} \).

  15. 15.

    Technically, (5.54) is written as a cross-correlation and not a convolution. However, since the weights w are variables to be learned, and all quantities are real-valued, the distinction is not important.

  16. 16.

    A key distinction between recurrent networks and HMMs is that the “state space” in a recurrent network is continuous, i.e., \( h_{t} \in \mathbb{R}^{d_{i}} \).

  17. 17.

    The presentation of Graves [57] differs slightly in its inclusion of “peephole” connections [51]. We omit these connections here for clarity of presentation, and because recent studies have not demonstrated their efficacy [60].

  18. 18.

    Some authors define the BRNN output (5.62) as a non-linear transformation of the concatenated state vectors [55]. This formulation is equivalent to (5.62) followed by a one-dimensional convolutional layer with a receptive field n i = 1, so we opt for the simpler definition here.

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  1. Center for Data Science, New York University, 60 5th Ave., New York, NY, 10003, USA

    Brian McFee

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  1. Laboratory of Signal Processing, Tampere University of Technology, Tampere, Finland

    Tuomas Virtanen

  2. Centre for Vision, Speech and Signal Processing, University of Surrey, Surrey, United Kingdom

    Mark D. Plumbley

  3. Google Inc., New York, New York, USA

    Dan Ellis

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McFee, B. (2018). Statistical Methods for Scene and Event Classification. In: Virtanen, T., Plumbley, M., Ellis, D. (eds) Computational Analysis of Sound Scenes and Events. Springer, Cham. https://doi.org/10.1007/978-3-319-63450-0_5

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