# Outlier Detection for Multivariate Time Series Using Dynamic Bayesian Networks

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background

#### 2.1. Bayesian Networks

#### 2.2. Dynamic Bayesian Networks

**Definition**

**1**

**Definition**

**2**

**Definition**

**3**

## 3. Methods

#### 3.1. Pre-Processing

Algorithm 1 Data Pre-Processing |

Input: A MTS dataset D of n variables along T instants; an alphabet size ${r}_{i}$ for each attribute ${X}_{i}\left[t\right]$, $1\le i\le n$; desired length $w\ll T$ of the resulting MTS.Output: The set of input MTS discretized.
1: procedure SAX(D,${r}_{i}$ for all i,w)2: for each subject h in D do3: for each TS ${\left\{{x}_{i}^{h}\left[t\right]\right\}}_{0\le t\le T}$, with $1\le i\le n$ do4: for each t, with $0\le t\le T$ do5: Normhi ^{h}_{i}[t] $\leftarrow z\_Norm\left({x}_{i}^{h}\left[t\right]\right)$ ▹Normalization6: function PAA($Nor{m}_{i}^{h},w$) ▹Dimensionality reduction7: $k\leftarrow 0$ 8: Partition the $Nor{m}_{i}^{h}$ in contiguous blocks of size $T/w$ 9: for each block $B\phantom{\rule{-0.166667em}{0ex}}{L}_{k}$ do10: ${\widehat{x}}_{i}^{h}\left[k\right]\leftarrow (w/T){\sum}_{t\in B\phantom{\rule{-0.166667em}{0ex}}{L}_{k}}Nor{m}_{i}^{h}\left[t\right]$ ▹ Compressed slices 11: $k\leftarrow k+1$ 12: function Discretization (${\widehat{x}}_{i}^{h}\left[k\right],{r}_{i}$) ▹ Symbolic discretization13: $\beta \leftarrow SegmentGaussianDistrib($${r}_{i}$) 14: for each value $val$ in ${\widehat{x}}_{i}^{h}\left[k\right]$ do15: Discrete ^{h}_{i}[k] $\leftarrow ToSymbolic(val,\beta )$16: $return\left(Discrete\right)$ ▹ Return discretized MTS dataset |

#### 3.2. Modeling

Algorithm 2 Optimal Non-Stationary m-Order Markov tDBN Learning |

Input: A set of input MTS discretized over w time slices; the Markov lag m; the maximum number of parents p from preceding time slices.Output: A tree-augmented DBN structure.1: procedure Tree-augmented DBN(MTS,m,p)2: for each transition $\{t-m,\dots ,t-1\}\to t$ do3: Build a complete directed graph in $\mathbf{X}\left[t\right]$ 4: Calculate the weight of all edges and the optimal set of $p+1$ parents 5: Apply a maximum branching algorithm 6: Extract transition $t-m\to t$ network and the optimal set of parents 7: Collect transition networks ${B}_{t-m}^{t}$ to obtain a tDBN structure |

#### 3.3. Scoring

**Definition**

**4**

Algorithm 3 Transition Outlier Detection |

Input: A tDBN storing conditional probabilities for each transition network ${B}_{t-m}^{t}$, a (discretized) MTS dataset D, and a threshold $thr$ to discern abnormality.Output: The set of anomalous transitions $t-m\to t$ with scores below $thr$.1: procedure2: for each time slice t do3: for each subject $h\in \mathcal{H}$ do4: function Scoring(${D}_{t-m:t}^{h},{B}_{t-m}^{t},t$)5: for each variable ${X}_{i}\left[t\right]$ do6: Π _{Xi}[t]$\leftarrow GetParents({X}_{i}\left[t\right],{B}_{t-m}^{t})$7: w ^{h}_{i}[t]$\leftarrow GetParentsConfig({\mathsf{\Pi}}_{{X}_{i}\left[t\right]},{D}_{t-m:t}^{h})$8: p ^{h}_{i}$\leftarrow GetProbability({x}_{i}^{h}\left[t\right],{w}_{i}^{h}\left[t\right],{B}_{t-m}^{t})$9: P ^{h}_{i}$\leftarrow (1-{r}_{i}\xb7{y}_{min}){p}_{i}^{h}+{y}_{min}$ ▹ Probability smoothing10: s ^{h}_{t−m:t}$\leftarrow {\sum}_{i=1}^{n}log{P}_{i}^{h}$ ▹ Transition score11: if ${s}_{t-m:t}^{h}<thr$ then12: outliers $\leftarrow outliers$.append $\left({D}_{t-m:t}^{h}\right)$ |

#### 3.4. Parameter Tuning

#### 3.5. Score-Analysis

#### 3.5.1. Tukey’s Strategy

#### 3.5.2. Gaussian Mixture Model

## 4. Experimental Results

#### 4.1. Simulated Data

#### 4.1.1. Tukey’s Score-Analysis

#### 4.1.2. Gaussian Mixture Model

#### 4.1.3. Comparison between GMM and Tukey’s Score-Analysis

#### 4.1.4. Comparison with Probabilistic Suffix Trees

#### 4.2. ECG

#### 4.3. Mortality

#### 4.4. Pen-Digits

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Example of a stationary first-order Markov DBN. On the left, the prior network ${B}^{0}$, for $t=0$, and on the right, the transition network ${B}_{t-1}^{t}$ over slices $t-1$ and t, for all $t\ge 1$.

**Figure 2.**Scheme of the proposed outlier detection approach comprised of four phases. Datasets formed by MTS data can be directly applied to the modeling phase when discrete; otherwise, the pre-processing phase is applied before modeling. Discrete data is delivered to the modeling phase along with parameters p, m, and s of the DBN to be modeled. Afterward, a sliding window algorithm outputs a score distribution for the data (scoring entire MTS, called subjects, or only portions of it, called transitions, depending on the user’s choice). The score-analysis phase considers two distinct strategies providing thus two possible routes for outlier disclosure.

**Figure 3.**Transition networks of stationary first-order DBNs ($m=1$). The network (

**a**) on the left represents the transition network of DBN A which generates normal subjects. Networks (

**b**,

**c**) represent DBN B and C, respectively, which generate anomalous subjects. Dashed connections represent links which are removed with respect to the normal network (

**a**), while red links symbolize added dependencies. Solid black edges are connections which are common with respect to (

**a**).

**Figure 4.**Comparison between GMM and Tukey’s score-analysis ${\mathrm{F}}_{1}$ scores for multiple outlier ratios. Each value is an average of all 15 trials performed for each outlier ratio.

**Figure 5.**Subject outlierness using METEOR (

**a**) and PST approach (

**b**) for a same experiment of a dataset of 10,000 subjects ($N=10,000$) with 20% anomalies generated by model C. Histograms display thresholds using both score-analysis strategies. Scores below the threshold are classified as abnormal (in red) while the rest are classified as normal (in green), being the presented color representation for the Tukey’s thresholds.

**Figure 6.**Mean and standard deviation of normalized ECG variables along time using a SAX alphabet ${r}_{i}=5$ for $i=1,2$.

**Figure 7.**ECG transitions arranged by subject. A non-stationary second-order tDBN ($m=2$) model with inter-slice connectivity ($p=1$) is used together with Tukey’s score-analysis. Flipped subjects are associated to the highest subject ids. Data is discretized using SAX with an alphabet of 5 symbols (${r}_{i}=5$ for all i). Transitions displayed in red are classified as abnormal while in green are classified as normal.

**Figure 8.**Normalized values of variables ${X}_{i\in 1,\dots ,6}$ representing France’s mortality rates of males with ages 10, 20, 30, 40, 60 and 80, respectively, from 1841 to 1987. Each time stamp represents a year. Data is discretized with a SAX alphabet ${r}_{i}=5$ for all i.

**Figure 9.**Transition outlierness for mortality datasets of 5 (

**a**) and 6 (

**b**) variables using a third-order tDBN ($m=3$) with one inter-slice connectivity per node ($p=1$). Dataset (

**a**) is comprised by 5 variables ($n=5$) representing mortality rates of males with ages 20, 30, 40, 60 and 80. Dataset (

**b**) includes the same variables as (

**a**) with the addition of a variable representing the mortality rate of males aged 10 ($n=6$). Transitions are arranged by year and classified as anomalous (red) and normal (green). Major wars and epidemics which affected France in the selected years are exhibited.

Model B | Model C | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

${\mathit{P}}_{\mathit{O}}$ | $\mathit{N}$ | PPV | TPR | ACC | ${\mathbf{F}}_{\mathbf{1}}$ | $\mathit{N}$ | PPV | TPR | ACC | ${\mathbf{F}}_{\mathbf{1}}$ |

100 | 0.88 | 0.70 | 0.98 | 0.78 | 100 | 0.89 | 0.73 | 0.98 | 0.80 | |

5 | 1000 | 0.93 | 0.96 | 0.99 | 0.94 | 1000 | 0.91 | 0.98 | 0.99 | 0.94 |

10,000 | 0.95 | 0.98 | 0.99 | 0.96 | 10,000 | 0.94 | 1.00 | 0.99 | 0.97 | |

100 | 0.96 | 0.38 | 0.94 | 0.54 | 100 | 0.89 | 0.73 | 0.97 | 0.80 | |

10 | 1000 | 0.99 | 0.87 | 0.99 | 0.93 | 1000 | 0.97 | 0.87 | 0.98 | 0.92 |

10,000 | 0.99 | 0.91 | 0.99 | 0.95 | 10,000 | 0.99 | 0.87 | 0.98 | 0.93 | |

100 | 1.00 | 0.19 | 0.83 | 0.32 | 100 | 0.90 | 0.22 | 0.84 | 0.35 | |

20 | 1000 | 1.00 | 0.20 | 0.84 | 0.33 | 1000 | 1.00 | 0.37 | 0.87 | 0.54 |

10,000 | 1.00 | 0.16 | 0.83 | 0.28 | 10,000 | 1.00 | 0.29 | 0.86 | 0.45 |

Model B | Model C | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

${\mathit{P}}_{\mathit{O}}$ | $\mathit{N}$ | PPV | TPR | ACC | ${\mathbf{F}}_{\mathbf{1}}$ | $\mathit{N}$ | PPV | TPR | ACC | ${\mathbf{F}}_{\mathbf{1}}$ |

100 | 0.82 | 0.70 | 0.98 | 0.76 | 100 | 0.64 | 1.00 | 0.96 | 0.78 | |

5 | 1000 | 0.91 | 0.97 | 0.99 | 0.94 | 1000 | 0.86 | 0.99 | 0.99 | 0.92 |

10,000 | 0.95 | 0.98 | 0.99 | 0.96 | 10,000 | 0.98 | 1.00 | 0.99 | 0.99 | |

100 | 0.77 | 0.68 | 0.93 | 0.72 | 100 | 0.92 | 0.78 | 0.97 | 0.84 | |

10 | 1000 | 0.94 | 0.96 | 0.99 | 0.95 | 1000 | 0.89 | 0.97 | 0.98 | 0.93 |

10,000 | 0.91 | 0.98 | 0.99 | 0.94 | 10,000 | 0.93 | 0.96 | 0.99 | 0.95 | |

100 | 0.66 | 0.49 | 0.85 | 0.56 | 100 | 0.75 | 0.58 | 0.88 | 0.65 | |

20 | 1000 | 0.86 | 0.89 | 0.94 | 0.87 | 1000 | 0.91 | 0.92 | 0.96 | 0.92 |

10,000 | 0.86 | 0.94 | 0.96 | 0.90 | 10,000 | 0.93 | 0.94 | 0.97 | 0.94 |

**Table 3.**PST results using Tukey and GMM strategies on simulated data for experiments with $N=$ 10,000.

Tukey’s Strategy | ||||||||
---|---|---|---|---|---|---|---|---|

Model B | Model C | |||||||

${\mathit{P}}_{\mathit{O}}$ | PPV | TPR | ACC | ${\mathbf{F}}_{\mathbf{1}}$ | PPV | TPR | ACC | ${\mathbf{F}}_{\mathbf{1}}$ |

5 | 0.96 | 0.73 | 0.98 | 0.83 | 0.96 | 0.94 | 0.99 | 0.95 |

10 | 0.70 | 0.02 | 0.90 | 0.04 | 0.98 | 0.39 | 0.94 | 0.56 |

20 | 0.42 | 0.00 | 0.80 | 0.00 | 1.00 | 0.03 | 0.81 | 0.06 |

GMM Strategy | ||||||||

Model B | Model C | |||||||

${\mathit{P}}_{\mathit{O}}$ | PPV | TPR | ACC | ${\mathbf{F}}_{\mathbf{1}}$ | PPV | TPR | ACC | ${\mathbf{F}}_{\mathbf{1}}$ |

5 | 0.86 | 0.88 | 0.99 | 0.87 | 0.94 | 0.95 | 0.99 | 0.94 |

10 | 0.20 | 0.87 | 0.65 | 0.33 | 0.88 | 0.68 | 0.96 | 0.77 |

20 | 0.25 | 0.67 | 0.53 | 0.36 | 0.763 | 0.883 | 0.92 | 0.82 |

Experiment | TP | FP | TN | FN | PPV | TPR | ACC | ${\mathbf{F}}_{1}$ |
---|---|---|---|---|---|---|---|---|

${D}_{7}$ | 24 | 41 | 1102 | 106 | 0.37 | 0.18 | 0.88 | 0.25 |

${D}_{8}$ | 98 | 45 | 1098 | 32 | 0.69 | 0.75 | 0.94 | 0.72 |

${D}_{9}$ | 90 | 42 | 1101 | 40 | 0.68 | 0.69 | 0.94 | 0.69 |

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**MDPI and ACS Style**

Serras, J.L.; Vinga, S.; Carvalho, A.M.
Outlier Detection for Multivariate Time Series Using Dynamic Bayesian Networks. *Appl. Sci.* **2021**, *11*, 1955.
https://doi.org/10.3390/app11041955

**AMA Style**

Serras JL, Vinga S, Carvalho AM.
Outlier Detection for Multivariate Time Series Using Dynamic Bayesian Networks. *Applied Sciences*. 2021; 11(4):1955.
https://doi.org/10.3390/app11041955

**Chicago/Turabian Style**

Serras, Jorge L., Susana Vinga, and Alexandra M. Carvalho.
2021. "Outlier Detection for Multivariate Time Series Using Dynamic Bayesian Networks" *Applied Sciences* 11, no. 4: 1955.
https://doi.org/10.3390/app11041955