# Sensor Compromise Detection in Multiple-Target Tracking Systems

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## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. Problem Statement

#### 2.2. Multiple-Target Estimation with the PHD Filter

#### 2.3. Multiple-Sensor PHD Filter

#### 2.4. Target Detection from the SMC–PHD Algorithm

## 3. Methods

#### 3.1. Preliminaries

- Modified observation of a given target: The sensor noise does not conform to the assumed distribution. This can be, for example, an added bias on the observations or an increased covariance in the noise distribution.
- Modified probability of target detection: For instance, some targets can be omitted from the sensor observations.
- Modified false alarm rate: This can include an increase in the clutter intensity, duplicate reports for a given target, and so on.

**Assumption**

**1.**

**Assumption**

**2.**

**Assumption**

**3.**

#### 3.2. OSPA Metric

#### 3.3. Anomaly Detection: Single-Sensor Case

**Proposition**

**1.**

**Proof.**

#### 3.4. Anomaly Detection for Multiple-Sensor Systems

**Assumption**

**4.**

#### 3.5. Algorithm for Multiple-Sensor Anomaly Detection

- Data collection. Gather data from all sensors during nominal operation. This requires a trusted environment. This data is then used to compute ${\mathrm{\Lambda}}_{k}^{[i]}$ and ${\mathrm{{\rm Y}}}_{k}^{[i]}$ for all k in the recorded measurements.
- Nonparametric statistical test. During a target tracking task, keep a history of measurements for every sensor using a fixed horizon; that is, collect ${\overline{\mathrm{\Lambda}}}_{k}^{[i]},{\overline{\mathrm{{\rm Y}}}}_{k}^{[i]}$ for a lag l. At every time step k, perform a nonparametric statistical test between the history of measurements and the trusted data for each sensor, and save the p-value of the test.
- Bayesian inference for anomaly detection. Denote the trustworthiness of each sensor by the indicator function:$${t}_{k}^{[i]}=\left(\right)open="\{"\; close>\begin{array}{cc}0\hfill & \mathrm{Sensor}\phantom{\rule{4.pt}{0ex}}i\phantom{\rule{4.pt}{0ex}}\mathrm{cannot}\phantom{\rule{4.pt}{0ex}}\mathrm{be}\phantom{\rule{4.pt}{0ex}}\mathrm{trusted}\phantom{\rule{4.pt}{0ex}}\mathrm{at}\phantom{\rule{4.pt}{0ex}}\mathrm{time}\phantom{\rule{4.pt}{0ex}}k\hfill \\ 1\hfill & \mathrm{Sensor}\phantom{\rule{4.pt}{0ex}}i\phantom{\rule{4.pt}{0ex}}\mathrm{is}\phantom{\rule{4.pt}{0ex}}\mathrm{fully}\phantom{\rule{4.pt}{0ex}}\mathrm{reliable}\phantom{\rule{4.pt}{0ex}}\mathrm{at}\phantom{\rule{4.pt}{0ex}}\mathrm{time}\phantom{\rule{4.pt}{0ex}}k\hfill \end{array}$$$${c}_{k}^{[i]}=\left(\right)open="\{"\; close>\begin{array}{cc}0\hfill & \mathrm{Sensor}\phantom{\rule{4.pt}{0ex}}i\phantom{\rule{4.pt}{0ex}}\mathrm{reports}\phantom{\rule{4.pt}{0ex}}\mathrm{an}\phantom{\rule{4.pt}{0ex}}\mathrm{anomalous}\phantom{\rule{4.pt}{0ex}}\mathrm{{\rm Y}}\phantom{\rule{4.pt}{0ex}}\mathrm{at}\phantom{\rule{4.pt}{0ex}}\mathrm{time}\phantom{\rule{4.pt}{0ex}}k\hfill \\ 1\hfill & \mathrm{Sensor}\phantom{\rule{4.pt}{0ex}}i\phantom{\rule{4.pt}{0ex}}\mathrm{has}\phantom{\rule{4.pt}{0ex}}\mathrm{a}\phantom{\rule{4.pt}{0ex}}\mathrm{nominal}\phantom{\rule{4.pt}{0ex}}{\mathrm{{\rm Y}}}_{k}\hfill \end{array}$$$${e}_{k}^{[i]}=\left(\right)open="\{"\; close>\begin{array}{cc}0\hfill & \mathrm{Sensor}\phantom{\rule{4.pt}{0ex}}i\phantom{\rule{4.pt}{0ex}}\mathrm{reports}\phantom{\rule{4.pt}{0ex}}\mathrm{an}\phantom{\rule{4.pt}{0ex}}\mathrm{anomalous}\phantom{\rule{4.pt}{0ex}}\mathrm{\Lambda}\phantom{\rule{4.pt}{0ex}}\mathrm{at}\phantom{\rule{4.pt}{0ex}}\mathrm{time}\phantom{\rule{4.pt}{0ex}}k\hfill \\ 1\hfill & \mathrm{Sensor}\phantom{\rule{4.pt}{0ex}}i\phantom{\rule{4.pt}{0ex}}\mathrm{has}\phantom{\rule{4.pt}{0ex}}\mathrm{a}\phantom{\rule{4.pt}{0ex}}\mathrm{nominal}\phantom{\rule{4.pt}{0ex}}{\mathrm{\Lambda}}_{k}\hfill \end{array}$$$$\begin{array}{c}p\left(\right)open="("\; close=")">{t}_{k}^{[i]}|{e}_{k-1}^{[i]},{c}_{k-1}^{[i]},{t}_{k-1}^{[i]}=\sum _{j}f\left(\right)open="("\; close=")">{t}_{k}^{[i]}|{t}_{k-1}^{[i]}=j& p\left(\right)open="("\; close=")">{t}_{k-1}^{[i]}=j|{e}_{1:k-1}^{[i]},{c}_{1:k-1}^{[i]}\end{array}$$$$\begin{array}{c}p\left(\right)open="("\; close=")">{t}_{k}^{[i]}|{e}_{k}^{[i]},{c}_{k}^{[i]}=\frac{p\left(\right)open="("\; close=")">{e}_{k}^{[i]}|{t}_{k}^{[i]}}{p}p\left(\right)open="("\; close=")">{t}_{k}^{[i]}|{e}_{k-1}^{[i]},{c}_{k-1}^{[i]},{t}_{k-1}^{[i]}\\ {\sum}_{j}p\left(\right)open="("\; close=")">{e}_{k}^{[i]}|{t}_{k}^{[i]}=jp\left(\right)open="("\; close=")">{c}_{k}^{[i]}|{e}_{k}^{[i]},{t}_{k}^{[i]}=j\end{array}$$

## 4. Results

#### 4.1. Coordinated Turn Model

#### 4.2. Simulated Attacks

- Zero-mean noise introduced.
- Mobile decoy added.
- Static decoy inserted.
- Bias added to the observation of one target.
- Multiple decoys added.
- A target surpressed.
- Bias added to all observations.

## 5. Discussion

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

BN | Bayesian network |

LTI | Linear time-invariant |

OSPA | Optimal sub-pattern assignment |

PHD | Probability hypothesis density |

SVM | Support vector machine |

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**Figure 4.**Ground truth of the target tracks. The $(\circ )$ markers indicate the start of the target tracks, and their ends are the $(\u25b5)$ markers. The $(\times )$ markers indicate the locations of the eight sensors.

**Figure 5.**Example of an unsuccessful attack. Several decoys were added to three sensors, out of eight. The cardinality of the detected target set is still correct, even in the presence of the false observations. The Bayesian scheme was able to identify the compromised sensors, while the SVM classifier incurred many false positives. (

**a**) Sensor compromise detection by Bayesian network (BN) and support vector machine (SVM), compared to ground truth; (

**b**) Tracks detected by the sequential Monte Carlo—probability hypothesis density (SMC—PHD) filter, with every sensor observation displayed; (

**c**) Optimal sub-pattern assignment (OSPA) metric, along with its location and cardinality error components. True and estimated cardinalities are also displayed.

**Figure 6.**A successful attack. An increased level of white noise was injected to three sensors, out of eight. The cardinality of the detected target set is incorrect for the time of the attack. Both the Bayesian scheme and the SVM classifier flagged many false positives. (

**a**) Sensor compromise detection by Bayesian network (BN) and support vector machine (SVM), compared to ground truth; (

**b**) Tracks detected by the sequential Monte Carlo—probability hypothesis density (SMC—PHD) filter, with every sensor observation displayed; (

**c**) Optimal sub-pattern assignment (OSPA) metric, along with its location and cardinality error components. True and estimated cardinalities are also displayed.

**Figure 7.**An attack on all sensors. No compromise detection method was able to identify the intrusion. (

**a**) Sensor compromise detection by Bayesian network (BN) and support vector machine (SVM), compared to ground truth; (

**b**) Tracks detected by the sequential Monte Carlo—probability hypothesis density (SMC—PHD) filter, with every sensor observation displayed; (

**c**) Optimal sub-pattern assignment (OSPA) metric, along with its location and cardinality error components. True and estimated cardinalities are also displayed.

Attack | F-Score Mode | BN Mean | BN Std | BN Best | BN Worst | SVM Mean | SVM Std | SVM Best | SVM Worst |
---|---|---|---|---|---|---|---|---|---|

1 | Per sensor | 0.4447 | 0.0711 | 0.5104 | 0.2555 | 0.3042 | 0.0720 | 0.4294 | 0.1582 |

1 | Global | 0.8534 | 0.0829 | 0.9618 | 0.6667 | 0.6190 | 0.1089 | 0.7826 | 0.3214 |

2 | Per sensor | 0.8547 | 0.0501 | 0.9174 | 0.7494 | 0.5160 | 0.0550 | 0.6008 | 0.3817 |

2 | Global | 0.8919 | 0.0618 | 0.9576 | 0.6986 | 0.8024 | 0.0944 | 0.9243 | 0.5472 |

3 | Per sensor | 0.8414 | 0.0625 | 0.9329 | 0.6667 | 0.6434 | 0.0948 | 0.7742 | 0.4393 |

3 | Global | 0.8741 | 0.0656 | 0.9600 | 0.7126 | 0.7931 | 0.0914 | 0.9200 | 0.5882 |

4 | Per sensor | 0.4843 | 0.0232 | 0.5180 | 0.4398 | 0.4552 | 0.0489 | 0.5125 | 0.3557 |

4 | Global | 0.8628 | 0.0597 | 0.9531 | 0.7423 | 0.8162 | 0.0998 | 0.9434 | 0.6133 |

5 | Per sensor | 0.8485 | 0.0547 | 0.9300 | 0.7550 | 0.4759 | 0.0568 | 0.5684 | 0.3623 |

5 | Global | 0.9051 | 0.0424 | 0.9627 | 0.8148 | 0.8077 | 0.0871 | 0.9339 | 0.6500 |

6 | Per sensor | 0.0355 | 0.0281 | 0.0724 | 0.0026 | 0.2126 | 0.0636 | 0.3232 | 0.0778 |

6 | Global | 0.8968 | 0.0449 | 0.9565 | 0.8000 | 0.8146 | 0.0764 | 0.9381 | 0.6389 |

7 | Per sensor | 0.4812 | 0.0233 | 0.5203 | 0.4279 | 0.4333 | 0.0383 | 0.4901 | 0.3587 |

7 | Global | 0.8758 | 0.0586 | 0.9573 | 0.7209 | 0.7842 | 0.0818 | 0.9239 | 0.6250 |

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**MDPI and ACS Style**

Ramirez-Paredes, J.-P.; Doucette, E.A.; Curtis, J.W.; Ayala-Ramirez, V.
Sensor Compromise Detection in Multiple-Target Tracking Systems. *Sensors* **2018**, *18*, 638.
https://doi.org/10.3390/s18020638

**AMA Style**

Ramirez-Paredes J-P, Doucette EA, Curtis JW, Ayala-Ramirez V.
Sensor Compromise Detection in Multiple-Target Tracking Systems. *Sensors*. 2018; 18(2):638.
https://doi.org/10.3390/s18020638

**Chicago/Turabian Style**

Ramirez-Paredes, Juan-Pablo, Emily A. Doucette, Jess W. Curtis, and Victor Ayala-Ramirez.
2018. "Sensor Compromise Detection in Multiple-Target Tracking Systems" *Sensors* 18, no. 2: 638.
https://doi.org/10.3390/s18020638