The effect of the particle filter is visible from the two contrasting plots of telemetry data from the robot taken from two runs within the same pen in Figure 3.9. The black line is a plot of the telemetry without a particle filter. Integration errors from the dead reckoning algorithm are introduced by the wheels slipping instead of producing useful torque. The accumulation of these errors produces a stereotypical “spirograph” effect, with the same shape being incrementally redrawn by an increasingly erroneous rotational offset.
A Wilcoxon test on the false positives was unable to reject the hypoth- esis that the data after 1 minute had a different median to the data after 2 minutes (confidence level of 95%). This doesn’t prove that the data has the same median, but suggests an improvement in the consistency. A similar Wilcoxon on the false negative data did reject the hypothesis that they had the same medians, suggesting that some changes in per- formance do occur. Performing the same statistical test for difference on the results from the experiments after 2 mins before and after the localisation changes, also rejects the hypothesis that they are the same (confidence at 95%). With the effects of localisation error reduced, it is
−1500 −1000 −500 0 500 1000 1500 −1500 −1000 −500 0 500 1000 1500 2000 b[, 1] b[, 2]
Figure 3.9: Telemetry Data With and Without the Particle Filter The black line is the telemetry without the particle filter. The “spirograph” effect of a similar shape being repeated, slightly offset is a classic sign of integration errors in the dead-reckoning system. The red lines overlap significantly more as the estimate is repeatedly corrected.
All of the false positive rates are better than the random classifier performance. Three out of five false negative rates are better than a random classifier performance. This allows the second hypothesis of the experiment to be accepted as three parametrisations have been found which outperform the random classifier. It is an intuitive result that there would be a trade-off between a system which errs on the side of caution, (resulting in a high false negative rate) and a system which is more susceptible to triggering (resulting in a higher false positive rate). An additional phenomena is also present in the DCA. As all cells receive the same input signals, cells with smaller migration thresholds will tend to migrate after observing a smaller amount of data than cells with larger migration thresholds. This means that small migration thresholds can lead to snap judgements based on noise or sudden changes in circum- stance and that large thresholds can have their classification coloured by old, sometimes out-of-date data. These effects mean that the migration threshold distribution must be selected very carefully.
These results support the concept that the DCA is a potentially useful algorithm for a robotic security system. Despite the data being unfiltered,
M15 M30 M60 M120 M240 0.0 0.1 0.2 0.3 0.4
False Positive and False Negative Rates
False Positive False Negative
FP of Random Classifier FN of Random Classifier
Figure 3.10: Error Rates from the Modified DCA Experiments The dotted lines indicate the performance of a random classi- fier. The vertical axis is measured in terms of a fraction of the total number of classifications made. The error bars show the minimum and maximum errors.
well as providing sufficient computer time for vital functions. Secondly, the importance of using a good localisation algorithm has been clearly shown. The performance of the DCA without the particle filter enabled was worse than a random classifier in terms of its false negative rate. The cause of this was that the mapping produced by the antigen gen- eration function was losing its correspondence to the robot’s location. Localisation algorithms are typically computationally expensive and the requirement to pre-load a map limits the adaptability of the system. However, a statistically detectable improvement was produced by adding the localised data.
This suggests several areas of further work. The first area concerns the DCA’s tolerance to noise. The reaction of the DCA to the unfiltered inputs seems to be tolerant to noise. The multiple time scales that the cells sample over, as a result of their varied migration thresholds, would seem to be influencing the way the DCA handles input data. As noise is a typically high-frequency signal, it is a valid hypothesis that the frequency of a signal within the input data has some effect on the weighting it receives in the decision-making process. Chapter 4 is an exploration of this hypothesis, utilising frequency-domain analysis of a single dendritic cell. Now that the performance of the DCA has been shown to be at least
established, equivalent system in order to gauge exactly how well it is performing. This is done through a theoretical analysis of the algorithm in Chapter 7.
3.7.1
Summary
In this chapter a new, optimised version of the DCA was designed and implemented on a mobile robot. The new version was able to produce results as the robot patrolled around an arena demonstrating that the system was both fast enough to prevent interference with the primary functions of the robot and produce results on-line instead of in a batch fashion. The parametrisation of the DCA was explored for a mock secu- rity scenario which featured a signature based metric for the PAMP sig- nal, a Danger signal that had a high false positive rate and an inhibitory Safe signal that prevented the system reacting to anomalies that had been identified as acceptable. It was found that with the correct parametrisa- tion, the DCA can at least outperform a purely random classifier at this task.
3.7.2
Contributions
Novel contributions provided by this chapter are:
• The use of an antigen multiplication function, as opposed to a static antigen multiplier.
• The conversion of a batch-processing version of the DCA into one suitable for use on a real-time system. The interested reader is di- rected to [52] which provides a full exploration of real-time methods for the DCA, including the technique used for this experiment. • The implementation of the abstract K variable, which makes pos-
“I don’t think necessity is the mother of invention. Invention, in my opinion, arises directly from idleness, possibly also from laziness - to save oneself trouble.” - Agatha Christie, An Autobiography (1977)
4.1
Introduction
The work in this chapter is based on [108].
Despite a large volume of research being carried out on the DCA, its behaviour in neither the time nor the frequency domain is well un- derstood. Results from Chapter 3 suggested that the algorithm treats signals of differing frequency differently, but the relationship between the algorithm’s input parameters and its behaviour in the frequency domain has never been characterised.
In this chapter a single dendritic cell will be modelled in the frequency domain as a digital filter. Section 4.2 demonstrates how the mathematics of the signal processing equations can be rephrased as a filter. Section 4.3 outlines an experiment to compare the results of the resulting model with the original algorithm and Section 4.4 presents the results of that experiment. Section 4.5 discusses the findings of the chapter and the limitations of the single-cell model.