Motivation: Fence-mounted perimeter intrusion detection systems

• Data obtained using fiber optic cables attached to a fence

Motivation: Fence-mounted perimeter intrusion detection systems

• Data obtained using fiber optic cables attached to a fence
• Intrusion events cause measurable changes in intensity, phase, wavelength or transit time of light in the fiber.

Motivation: Fence-mounted perimeter intrusion detection systems

• Data obtained using fiber optic cables attached to a fence
• Intrusion events cause measurable changes in intensity, phase, wavelength or transit time of light in the fiber.
• Aim: Find anomalous time series (the location of the intrusion event)

Motivation: Network intrusion detection systems

• Yahoo data breach in late 2014 --- world's largest ever cyber attack

Motivation: Network intrusion detection systems

• Yahoo data breach in late 2014 --- world's largest ever cyber attack
• Intrusion attacks cause measurable changes in times of logins, command executed during a single user session, number of password failures

Motivation: Network intrusion detection systems

• Yahoo data breach in late 2014 --- world's largest ever cyber attack
• Intrusion attacks cause measurable changes in times of logins, command executed during a single user session, number of password failures
• Aim: find anomalous time series (locate intrusion attacks)

Motivation

• All these applications generate millions or even billions of individual time series simultaneously

Motivation

• All these applications generate millions or even billions of individual time series simultaneously
• Research question: Finding anomalous time series within a large collection of time series

Motivation

• All these applications generate millions or even billions of individual time series simultaneously
• Research question: Finding anomalous time series within a large collection of time series
• Approaches to solving the problem of anomaly detection for temporal data :

Motivation

• All these applications generate millions or even billions of individual time series simultaneously
• Research question: Finding anomalous time series within a large collection of time series
• Approaches to solving the problem of anomaly detection for temporal data :

  Batch scenario:
  whole set of data is available, focus - complete events

  

Motivation

• All these applications generate millions or even billions of individual time series simultaneously
• Research question: Finding anomalous time series within a large collection of time series
• Approaches to solving the problem of anomaly detection for temporal data :

  Batch scenario:
  whole set of data is available, focus - complete events

  

  Data stream scenario: continuous, unbounded, flow at high speed, high volume

  

Automatic anomaly detection algorithm for streaming data is required:

• to give real-time support

Automatic anomaly detection algorithm for streaming data is required:

• to give real-time support

• to provide early detection of anomalies

Automatic anomaly detection algorithm for streaming data is required:

• to give real-time support

• to provide early detection of anomalies

• to learn and adapt to the changing environment automatically (concept drift)

Automatic anomaly detection algorithm for streaming data is required:

• to give real-time support

• to provide early detection of anomalies

• to learn and adapt to the changing environment automatically (concept drift)

• to deal with large amounts of data efficiently

Algorithm of the proposed framework

Aim

• To detect anomalous time series within a large collection of time series in a streaming data context

Algorithm of the proposed framework

Aim

• To detect anomalous time series within a large collection of time series in a streaming data context

Main Assumptions

Algorithm of the proposed framework

Aim

• To detect anomalous time series within a large collection of time series in a streaming data context

Main Assumptions

• Anomaly is a rare event which has a very low chance of occurrence with respect to the typical behavior of the system

Algorithm of the proposed framework

Aim

• To detect anomalous time series within a large collection of time series in a streaming data context

Main Assumptions

• Anomaly is a rare event which has a very low chance of occurrence with respect to the typical behavior of the system
• A representative data set of the system's typical behavior is available to define the model for the typical behavior of the system.

Algorithm of the proposed framework

Aim

• To detect anomalous time series within a large collection of time series in a streaming data context

Main Assumptions

• Anomaly is a rare event which has a very low chance of occurrence with respect to the typical behavior of the system
• A representative data set of the system's typical behavior is available to define the model for the typical behavior of the system.

Proposed Algorithm

Algorithm of the proposed framework

Aim

• To detect anomalous time series within a large collection of time series in a streaming data context

Main Assumptions

• Anomaly is a rare event which has a very low chance of occurrence with respect to the typical behavior of the system
• A representative data set of the system's typical behavior is available to define the model for the typical behavior of the system.

Proposed Algorithm

• Off-line Phase: Building a model of a system's typical behaviour; (similar to (Clifton, Hugueny & Tarassenko, 2011))

Algorithm of the proposed framework

Aim

• To detect anomalous time series within a large collection of time series in a streaming data context

Main Assumptions

• Anomaly is a rare event which has a very low chance of occurrence with respect to the typical behavior of the system
• A representative data set of the system's typical behavior is available to define the model for the typical behavior of the system.

Proposed Algorithm

• Off-line Phase: Building a model of a system's typical behaviour; (similar to (Clifton, Hugueny & Tarassenko, 2011))
• On-line Phases: Testing newly arrived data using the boundary

Feature Based Representation of Time series

Theorem 1: Fisher-Tippett theorem (Limit laws for maxima)

(Embrechts et al. (2013), p. 121)

Let $X=X_1,X_2,...,X_m$ be a sequence of independent and identically distributed random variables and $X_{max}=max\left(X\right)$. If there exist centering constant $d_m(\in R)$ and normalizing constant $c_m(>0)$, and some non-degenerate distribution function $H^{+}$ such that

then $H^{+}$ belongs to one of the following three distribution functions:

Extreme Value Distribution of the Probability Density Values (Clifton et al., 2011)

• Estimate the probability density function of the 2D PC space --> Kernel density estimation
• Draw a large number N of extremes from the estimated probability density function

Figure: Distribution of 1000 extremes generated from bivariate kernel density function with m=500

Extreme Value Distribution of Probability Density Values (Clifton et al., 2011)

• Define a $\Psi$-transform space, using the $\Psi$-transformation defined by

• $\Psi$-transform maps the density values back into space into which a Gumbel distribution can be fitted.

Figure: Distribution of transformed values

How it works?

How it works?

What Next?

• Explore more on feature extraction and feature selection methods to create a better feature space suitable for streaming data context.

What Next?

• Explore more on feature extraction and feature selection methods to create a better feature space suitable for streaming data context.

• Use other dimension reduction techniques such as multidimensional scaling analysis, random projection to see the effect on the performance of the proposed framework.

What Next?

• Explore more on feature extraction and feature selection methods to create a better feature space suitable for streaming data context.

• Use other dimension reduction techniques such as multidimensional scaling analysis, random projection to see the effect on the performance of the proposed framework.

• Do more experiments on density estimation methods to get a better tail estimation.

What Next?

• Explore more on feature extraction and feature selection methods to create a better feature space suitable for streaming data context.

• Use other dimension reduction techniques such as multidimensional scaling analysis, random projection to see the effect on the performance of the proposed framework.

• Do more experiments on density estimation methods to get a better tail estimation.

• Extend the algorithm to work with Multidimensional Multivariate Data Streams.

References

Images were taken:

• http://55ca7cd0-f8ac-0132-1185-705681baa5c1.s3-website-sa-east-1.amazonaws.com/defesanet/site/upload/news_image/2016/03/30157.jpg
• https://www.intel.co.uk/content/dam/www/public/emea/xe/en/images/it-managers/datacenter-corridor-16x9.jpg.rendition.intel.web.1280.720.jpg
• https://fibersensys.com/cache/mod_roksprocket/4d90594c170e9ec140017f0719ce2c98_350_900.jpg
• https://c1.staticflickr.com/8/7065/26946304530_cb30c23660_b.jpg

Main references

• Clifton, D. A., Hugueny, S., & Tarassenko, L. (2011). Novelty detection with multivariate extreme value statistics. Journal of signal processing systems, 65 (3), (pp. 371-389).
• Fulcher, B. D. (2012). Highly comparative time-series analysis. PhD thesis, University of Oxford.
• Hyndman, R. J., Wang, E., & Laptev, N. (2015). Large-scale unusual time series detection. In 2015 IEEE International Conference on Data Mining Workshop (ICDMW), (pp. 1616-1619). IEEE.

Acknowledgement

Statistical Society of Australia, Victorian Branch

• for offering financial support to attend the Young Statisticians Conference (YSC) 2017 in Coolangatta, QLD.