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Maritime abnormality detection using Gaussian processes

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Abstract

Novelty, or abnormality, detection aims to identify patterns within data streams that do not conform to expected behaviour. This paper introduces novelty detection techniques using a combination of Gaussian processes, extreme value theory and divergence measurement to identify anomalous behaviour in both streaming and batch data. The approach is tested on both synthetic and real data, showing itself to be effective in our primary application of maritime vessel track analysis.

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Acknowledgments

This work was funded by ISSG, Babcock Marine and Technology Division, Devonport Royal Dockyard. Ioannis Psorakis is funded from a grant via Microsoft Research, for which we are most grateful. This work was further supported by EPSRC project EP/I011587/1.

Author information

Authors and Affiliations

  1. ISSG, Babcock Marine and Technology Division, Devonport Royal Dockyard, Plymouth, PL1 4SG, UK

    Mark Smith

  2. Department of Engineering Science, University of Oxford, Oxford, UK

    Steven Reece, Stephen Roberts & Ioannis Psorakis

  3. Schlumberger Research, Cambridge, UK

    Iead Rezek

Authors
  1. Mark Smith
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  2. Steven Reece
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  3. Stephen Roberts
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  4. Ioannis Psorakis
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Corresponding author

Correspondence to Mark Smith.

Appendix

Appendix

Derivation of the Hellinger metric, Eq. 15.

The squared Hellinger distance is a measure of similarity between two probability distributions and is defined as

$$\begin{aligned} h^{2} \left( f \left( \varvec{x}\right) ,g \left( \varvec{x}\right) \right) =\frac{1}{2}\int \left( \sqrt{f \left( \varvec{x}\right) }-\sqrt{g \left( \varvec{x}\right) } \right) ^{2} \hbox {d}\varvec{x}, \end{aligned}$$
(16)

where \(f\left( \varvec{x}\right) \) and \(g\left( \varvec{x}\right) \) denote probability distributions. Equation 16 can alternatively be expressed as

$$\begin{aligned} h^{2}\left( f\left( \varvec{x}\right) ,g\left( \varvec{x}\right) \right) =1-\int \left( \sqrt{f \left( \varvec{x}\right) }\sqrt{g \left( \varvec{x}\right) } \right) \hbox {d}\varvec{x}. \end{aligned}$$

In the instance distributions \(f\left( \varvec{x}\right) \) and \(g\left( \varvec{x}\right) \) are multivariate Gaussian, the Hellinger distance would take the form

$$\begin{aligned}&h^{2} \left( f \left( \varvec{x};\varvec{\mu _{f}},\varvec{\Sigma _{f}}\right) ,g \left( \varvec{x};\varvec{\mu _{g}},\varvec{\Sigma _{g}}\right) \right) \\&\quad =1-\frac{1}{ \left( 2\pi \right) ^{\frac{n}{2}}\sqrt{|\varvec{\Sigma _{f}}|}}\frac{1}{ \left( 2\pi \right) ^{\frac{n}{2} }\sqrt{|\varvec{\Sigma _{g}}|}}\\&\qquad \times \int \sqrt{\exp \left( -\frac{1}{2} \left( \varvec{x}-\varvec{\mu _{f}}\right) ^{\top } \varvec{\Sigma _{f}}^{-1} \left( \varvec{x}-\varvec{\mu _{f}}\right) \right) }\\&\qquad \times \sqrt{\exp \left( \frac{1}{2} \left( \varvec{x}-\varvec{\mu _{g}}\right) ^{\top } \varvec{\Sigma _{g}}^{-1} \left( \varvec{x}-\varvec{\mu _{g}}\right) \right) }\hbox {d}\varvec{x}, \end{aligned}$$

The terms inside the exponents can also be combined and expressed in quadratic form, \((\varvec{x}-\varvec{\mu }^{*})^{\top }\varvec{C}^{-1}(\varvec{x}-\varvec{\mu }^{*})+\varvec{B}\), by making the following associations

$$\begin{aligned} \varvec{\mu }^{*}&= \left( \varvec{\Sigma _{f}}^{-1}+\varvec{\Sigma _{g}}^{-1}\right) ^{-1} \left( \varvec{\Sigma _{f}}^{-1}\varvec{\mu _{f}}+\varvec{\Sigma _{g}}^{-1} \varvec{\mu _{g}}\right) ,\\ \varvec{C}^{-1}&= \frac{1}{2}\varvec{\Sigma _{f}}^{-1}+\frac{1}{2}\varvec{\Sigma _{g}}^{-1},\\ \varvec{B}&= \left( \varvec{\mu _{f}}-\varvec{\mu _{g}}\right) ^{\top } \left( \varvec{\Sigma _{g}}+\varvec{\Sigma _{f}}\right) ^{-1} \frac{1}{2} \left( \varvec{\mu _{f}}-\varvec{\mu _{g}}\right) .\\ \end{aligned}$$

The integral can now be solved and the expression simplified

$$\begin{aligned}&h^{2} \left( f \left( \varvec{x}\right) ,g \left( \varvec{x}\right) \right) \\&\quad =1-\frac{|\frac{1}{2}\varvec{\Sigma _{f}}^{-1}+\frac{1}{2}\varvec{\Sigma _{g}}^{-1}|^{-\frac{1}{2}}}{|\varvec{\Sigma _{f}}|^{\frac{1}{4}}|\varvec{\Sigma _{g}}|^{\frac{1}{4}}}\\&\qquad \times \exp \left( -\frac{1}{4} \left( \varvec{\mu _{f}}-\varvec{\mu _{g}}\right) ^{\top } \left( \varvec{\Sigma _{g}}+\varvec{\Sigma _{f}}\right) ^{-1} \left( \varvec{\mu _{f}}-\varvec{\mu _{g}}\right) \right) . \end{aligned}$$

Under the assumption that both distributions have the same zero mean, \(\varvec{\mu }_f=\varvec{\mu }_g=\varvec{0}\), this can be further simplified

$$\begin{aligned}&h^{2} \left( f \left( \varvec{x};\varvec{\mu _{f}}= \varvec{0},\varvec{\Sigma _{f}}\right) ,g \left( \varvec{x};\varvec{\mu _{g}}=\varvec{0}, \varvec{\Sigma _{g}}\right) \right) \\&\quad =1-\frac{|\frac{1}{2}\varvec{\Sigma _{f}}^{-1}+\frac{1}{2} \varvec{\Sigma _{g}}^{-1}|^{-\frac{1}{2}}}{|\varvec{\Sigma _{f}}|^{\frac{1}{4}}| \varvec{\Sigma _{g}}|^{\frac{1}{4}}} \end{aligned}$$

To avoid the inverse covariances in this form, the fraction can be multiplied top and bottom by \(\left( |\varvec{\Sigma _{f}}||\varvec{\Sigma _{g}}|\right) ^{-\frac{1}{2}}\), in addition to the application of the determinant identity \(|\varvec{AB}|=|\varvec{A}||\varvec{B}|\), thus

$$\begin{aligned}&h^{2}\left( f \left( \varvec{x};\varvec{\mu _{f}}=\varvec{0},\varvec{\Sigma _{f}}\right) ,g \left( \varvec{x};\varvec{\mu _{g}}=\varvec{0},\varvec{\Sigma _{g}}\right) \right) \\&\quad =1-\frac{|\frac{1}{2}\varvec{\Sigma _{f}}+\frac{1}{2}\varvec{\Sigma _{g}}|^{-\frac{1}{2}}}{|\varvec{\Sigma _{f}}|^{-\frac{1}{4}}|\varvec{\Sigma _{g}}|^{-\frac{1}{4}}}\\&\quad =1-\sqrt{2}\frac{|\varvec{\Sigma _{f}}^{\frac{1}{4}}+\varvec{\Sigma _{g}}^{\frac{1}{4}}|}{|\varvec{\Sigma _{f}}+\varvec{\Sigma _{g}}|^{\frac{1}{2}}}\\ \end{aligned}$$

It can be noted, as a means of verifying the result, that by setting \(\varvec{\Sigma }=\sigma ^{2}\), the form is consistent with the Hellinger distance between two univariate Gaussian distributions when \(\mu _{f}=\mu _{g}=0\) namely

$$\begin{aligned} 1-\sqrt{\frac{2\sigma _{f}\sigma _{g}}{\sigma _{f}^{2}+\sigma _{g}^{2}}} \exp \left( -\frac{ \left( \mu _{f}-\mu _{g}\right) ^{2} }{4 \left( \sigma _{f}^{2}+\sigma _{g}^{2}\right) } \right) . \end{aligned}$$

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Smith, M., Reece, S., Roberts, S. et al. Maritime abnormality detection using Gaussian processes. Knowl Inf Syst 38, 717–741 (2014). https://doi.org/10.1007/s10115-013-0685-z

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