# TADILOF: Time Aware Density-Based Incremental Local Outlier Detection in Data Streams

^{*}

## Abstract

**:**

## 1. Introduction

- Continuous data points (usually infinite).
- Limited memory and limited computing power.
- Real time responses for processed data.

- Memory limitations constrain the amount of data that can be held in memory. We need to consider this for handling unbounded data stream environment.
- The state of the current data point as an outlier/inlier must be established before dealing with subsequent data points. Note that we do not have any information related to subsequent data points appearing in the data stream.
- Adding new data may induce new clusters.
- Limited computing power needs to be utilized before new data arrives in the data stream. Therefore, the algorithms need to be efficient in terms of execution time.

- We developed a novel algorithm to detect outliers in data streams. The proposed approach is capable of adapting the changes in variations of data over time.
- We developed an algorithm to calculate approximate LOF score in order to improve model performance.
- Extensive experiments using real-world datasets were performed to compare the performance of the proposed scheme with those of various state-of-the-art methods.
- The efficacy of the proposed scheme was demonstrated in a real-world pollution detection system using PM2.5 sensors.

## 2. Background and Related Work

#### 2.1. LOF and iLOF

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

#### 2.2. MiLOF

#### 2.3. DILOF

## 3. Proposed Method: TADILOF

Algorithm 1 TADILOF algorithm |

Input:
$DS$: A data stream $D=\{{d}_{1},{d}_{2},\dots ,{d}_{t},\dots \}$,Window size: W, Number of neighbor: K, Threshold: $\theta $, Step size: $\eta $, Regularization constant: $\lambda $, Maximum number of iteration: I Output: The set of outliers in streams1: $dataInMemory=\left\{\right\}$; 2: $outlierSet=\left\{\right\}$; 3: while a new data point ${d}_{t}$ is in stream do4: $dataInMemory$.add(${d}_{t}$) 5: $LO{F}_{k}\left({d}_{t}\right)$ = ODA(${d}_{t}$,$outlierSet$,$\theta $) 6: if $LO{F}_{k}\left({d}_{t}\right)>\theta $ then7: $outlierSet$.add(${d}_{t}$) 8: if $dataInMemory$.length $>W$ then9: $dataInMemory$=TADS($dataInMemory$,$\eta $,$\lambda $,I) 10: end while |

#### 3.1. Time Component

#### 3.2. Time-Aware Density Summarization (TADS)

Algorithm 2 Procedure TADS |

Input: set of data point in memory $X=\{{x}_{1},{x}_{2},\dots {x}_{W}\}$,Window size: W, Step size: $\eta $, Regularization constant: $\lambda $, Maximum number of iteration: I Output: summary set1: for each $datapoint\phantom{\rule{4pt}{0ex}}x\in X$ do2: if $LO{F}_{k}\left(x\right)<historicalLOF\left(x\right)$ then3: update LOF,LRD and meanDistance 4: end for5: $Y=\{{y}_{1},{y}_{2},\dots {y}_{W}\}$ 6: for each
$decision\phantom{\rule{4pt}{0ex}}variables\phantom{\rule{4pt}{0ex}}y\in Y$do7: y = 0.75 8: end for9: for i = 1:I do10: $\eta =\eta \ast 0.95$ 11: for n = 1:W do▹ Using objective function, calculate the score of each data point for selection in the summary set. 12: ${{y}_{n}}^{(i+1)}={{y}_{n}}^{\left(i\right)}-\eta \left\{t\_dif{f}_{n}+{\sum}_{x\in {C}_{K,n}}\frac{{p}_{k}\left(x\right)}{{v}_{k}\left(x\right)}-{e}^{LO{F}_{k}\left({x}_{n}\right)}{{\psi}_{0,1}}^{{}^{\prime}}\left({{y}_{n}}^{i}\right)+\lambda \left({\sum}_{n=1}^{W}{{y}_{n}}^{\left(i\right)}-\frac{3W}{4}\right)\right\}$ 13: end for14: end for15: Project Y into binary domain 16: for n=1:$\frac{3W}{4}$ do17: $Z\leftarrow Z\cup \left\{{x}_{n}\right\}$ 18: end for19: Return Z |

#### 3.3. LOF Score and ODA (Outlier Detection Using Approximate LOF)

Algorithm 3 Procedure of ODA |

Input:data point ${x}_{t}$set of data point in memory $X=\{{x}_{1},{x}_{2},\dots {x}_{t}\}$, threshold: $\theta $, set of detected outlier: $outlierSet$ Output: LOF score of ${x}_{t}$1: Using incremental LOF technique updates all reverse KNNs of ${x}_{t}$ 2: ${N}_{K}\left(x\right)$ = All KNNs of ${x}_{t}$ 3: for each
$neighbor$$n\in {N}_{K}\left(x\right)$do4: updating time stamp of n 5: end for6: Compute $LO{F}_{k}\left({x}_{t}\right)$ 7: if
$LO{F}_{k}\left({x}_{t}\right)>\theta $then8: Reference Point R= ${arg\; min}_{r\in {N}_{K}A}historicalLOF\left(r\right)\ast d(r,A)$ 9: Find the approximate reachability distance using Equation (12) 10: Find the approximate LRD of $\left({x}_{t}\right)$ using Equation (13) 11: Use historical LRD of R and historical LOF of R to find mean of LRD of potential neighbors by Equation (14) 12: Find the approximate LOF of $\left({x}_{t}\right)$ using Equation (15) 13: if approximate LOF of $\left({x}_{t}\right)>$ Threshold then14: $outlierSet$.add(${x}_{t}$) |

#### 3.4. Time and Space Complexity

## 4. Performance Evaluation

#### 4.1. Datasets

#### 4.2. Experiment Settings

#### 4.3. Experimental Results

#### 4.3.1. AUC, Execution Time, and Memory Usage

#### 4.3.2. Precision, Recall, and F1 Score

#### 4.4. Skipping Scheme for a Sequence of Outliers

## 5. PM2.5 Sensors Case Study

#### Monitoring a PM2.5 Pollution Event

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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Dataset | # Data Points | # Dimensions | # Outlier Data Points | Need to Shuffle |
---|---|---|---|---|

Annthyroid | 7200 | 6 | 534 | false |

Cardio | 1831 | 21 | 176 | true |

HTTP (KDD Cup 99) | 567,498 | 3 | 2211 | false |

Letter Recognition | 1600 | 32 | 100 | true |

Mnist | 7603 | 100 | 700 | true |

Musk | 3062 | 166 | 97 | true |

Pendigits | 6870 | 16 | 156 | false |

Satellite | 6435 | 36 | 2036 | false |

SMTP (KDD Cup 99) | 95,156 | 3 | 30 | false |

Vowels | 1456 | 12 | 50 | true |

Window Size | Precision | Recall | F1 Score | ||||||
---|---|---|---|---|---|---|---|---|---|

DILOF | TADILOF | MILOF | DILOF | TADILOF | MILOF | DILOF | TADILOF | MILOF | |

100 | 0.259074 | 0.224178 | 0.2289622 | 0.350187 | 0.383895 | 0.3506741 | 0.188322 | 0.198476 | 0.1945009 |

120 | 0.264844 | 0.222732 | 0.2331385 | 0.355993 | 0.392697 | 0.3582022 | 0.191396 | 0.200518 | 0.1975793 |

140 | 0.259869 | 0.213771 | 0.2369482 | 0.367790 | 0.404307 | 0.3630711 | 0.195014 | 0.201042 | 0.2018931 |

160 | 0.257486 | 0.218562 | 0.2381993 | 0.378652 | 0.415730 | 0.3679961 | 0.197863 | 0.206054 | 0.2023676 |

180 | 0.258819 | 0.217542 | 0.2464307 | 0.375094 | 0.418352 | 0.3726779 | 0.196032 | 0.207163 | 0.2043856 |

200 | 0.264608 | 0.218433 | 0.2433799 | 0.380899 | 0.426779 | 0.3750937 | 0.199770 | 0.210723 | 0.2032031 |

Window Size | Precision | Recall | F1 Score | ||||||
---|---|---|---|---|---|---|---|---|---|

DILOF | TADILOF | MILOF | DILOF | TADILOF | MILOF | DILOF | TADILOF | MILOF | |

100 | 0.3467338 | 0.3693657 | 0.3151908 | 0.3914205 | 0.4547727 | 0.3050568 | 0.2156009 | 0.2700938 | 0.1910918 |

120 | 0.3381342 | 0.3508179 | 0.3284806 | 0.3751136 | 0.4397159 | 0.3127273 | 0.2011454 | 0.2549688 | 0.1956810 |

140 | 0.3218308 | 0.3431242 | 0.3028065 | 0.3655682 | 0.4323864 | 0.2982955 | 0.1903846 | 0.2475356 | 0.1816338 |

160 | 0.3151459 | 0.3354626 | 0.3063062 | 0.3569886 | 0.4157387 | 0.3037500 | 0.1832554 | 0.2353132 | 0.1818394 |

180 | 0.3209461 | 0.3262367 | 0.3021858 | 0.3512500 | 0.4147726 | 0.3022158 | 0.1781800 | 0.2318135 | 0.1784863 |

200 | 0.3120879 | 0.3206106 | 0.2919695 | 0.3422159 | 0.4043182 | 0.2994319 | 0.1706907 | 0.2229609 | 0.1725868 |

Window Size | Precision | Recall | F1 Score | ||||||
---|---|---|---|---|---|---|---|---|---|

DILOF | TADILOF | MILOF | DILOF | TADILOF | MILOF | DILOF | TADILOF | MILOF | |

100 | 0.12697568 | 0.11241224 | 0.2220374 | 0.2059 | 0.2308 | 0.2593 | 0.06782202 | 0.08080138 | 0.1311881 |

120 | 0.14821722 | 0.16318930 | 0.2436584 | 0.2139 | 0.2443 | 0.2616 | 0.07340590 | 0.09222663 | 0.1351141 |

140 | 0.15472405 | 0.15568830 | 0.2298457 | 0.2193 | 0.2517 | 0.2618 | 0.07528581 | 0.09592009 | 0.1339924 |

160 | 0.17106840 | 0.17378370 | 0.2574958 | 0.2271 | 0.2625 | 0.2663 | 0.08395074 | 0.10338267 | 0.1392814 |

180 | 0.19773730 | 0.20144530 | 0.2839139 | 0.2335 | 0.2706 | 0.2718 | 0.08891001 | 0.11030273 | 0.1452346 |

200 | 0.20078190 | 0.19852930 | 0.2843155 | 0.2375 | 0.2732 | 0.2696 | 0.09236163 | 0.11257103 | 0.1431587 |

Window Size | Precision | Recall | F1 Score | ||||||
---|---|---|---|---|---|---|---|---|---|

DILOF | TADILOF | MILOF | DILOF | TADILOF | MILOF | DILOF | TADILOF | MILOF | |

100 | 0.221380000 | 0.191549667 | 0.240385667 | 0.209047667 | 0.243714000 | 0.241381000 | 0.074075133 | 0.107531667 | 0.135068000 |

120 | 0.200322333 | 0.272608333 | 0.254099000 | 0.212190333 | 0.248285667 | 0.242285667 | 0.076509233 | 0.112503333 | 0.138804667 |

140 | 0.198687667 | 0.260456000 | 0.292730000 | 0.216428333 | 0.257047667 | 0.249047667 | 0.080405100 | 0.121062667 | 0.140650667 |

160 | 0.228525333 | 0.293933667 | 0.301560333 | 0.217190333 | 0.262143000 | 0.252381000 | 0.080920167 | 0.125967000 | 0.144668000 |

180 | 0.177270000 | 0.281288667 | 0.307995000 | 0.219428667 | 0.265905000 | 0.257190333 | 0.083189067 | 0.127457667 | 0.145774667 |

200 | 0.186638333 | 0.297026333 | 0.298204333 | 0.221857000 | 0.270571333 | 0.257428333 | 0.084071200 | 0.133201333 | 0.147606667 |

Window Size | Precision | Recall | F1 Score | ||||||
---|---|---|---|---|---|---|---|---|---|

DILOF | TADILOF | MILOF | DILOF | TADILOF | MILOF | DILOF | TADILOF | MILOF | |

100 | 0.4421690 | 0.4141334 | 0.4397151 | 0.3313403 | 0.5407216 | 0.2925772 | 0.2013681 | 0.3326063 | 0.1927423 |

120 | 0.4083079 | 0.4027774 | 0.4104153 | 0.2854639 | 0.4829896 | 0.2662887 | 0.1614765 | 0.2968274 | 0.1652040 |

140 | 0.3923583 | 0.3881677 | 0.4092076 | 0.2637113 | 0.4541238 | 0.2397939 | 0.1438728 | 0.2802120 | 0.1452295 |

160 | 0.3857042 | 0.3906538 | 0.3659905 | 0.2461858 | 0.4086599 | 0.2360825 | 0.1276172 | 0.2505055 | 0.1363073 |

180 | 0.4097445 | 0.3819757 | 0.3576605 | 0.2322681 | 0.3759795 | 0.2064948 | 0.1189272 | 0.2324761 | 0.1081695 |

200 | 0.3928690 | 0.3710396 | 0.3543731 | 0.2198969 | 0.3363918 | 0.1968042 | 0.1107008 | 0.2034848 | 0.1058940 |

Window Size | Precision | Recall | F1 Score | ||||||
---|---|---|---|---|---|---|---|---|---|

DILOF | TADILOF | MILOF | DILOF | TADILOF | MILOF | DILOF | TADILOF | MILOF | |

100 | 0.0540172 | 0.0955094 | 0.10309758 | 0.312179 | 0.445513 | 0.3918589 | 0.0699342 | 0.103071 | 0.11157027 |

120 | 0.0517353 | 0.1142970 | 0.08978204 | 0.322436 | 0.483333 | 0.3849999 | 0.0718471 | 0.111540 | 0.10553305 |

140 | 0.0582843 | 0.0868875 | 0.08765809 | 0.331410 | 0.481410 | 0.3944872 | 0.0731726 | 0.108052 | 0.10599671 |

160 | 0.0553464 | 0.0676196 | 0.07356239 | 0.330128 | 0.485897 | 0.3857051 | 0.0716655 | 0.104400 | 0.09873375 |

180 | 0.0429529 | 0.0734877 | 0.07456543 | 0.314103 | 0.478205 | 0.3844233 | 0.0598091 | 0.105238 | 0.09594913 |

200 | 0.0500194 | 0.0743026 | 0.08276458 | 0.328205 | 0.484615 | 0.3871795 | 0.0667915 | 0.102561 | 0.09896692 |

Window Size | Precision | Recall | F1 Score | ||||||
---|---|---|---|---|---|---|---|---|---|

DILOF | TADILOF | MILOF | DILOF | TADILOF | MILOF | DILOF | TADILOF | MILOF | |

100 | 0.486230 | 0.488270 | 0.4720359 | 0.256925 | 0.333792 | 0.2466356 | 0.229341 | 0.303750 | 0.2279936 |

120 | 0.498198 | 0.496403 | 0.4636664 | 0.257122 | 0.341945 | 0.2488359 | 0.228337 | 0.307481 | 0.2281764 |

140 | 0.481029 | 0.494004 | 0.4694753 | 0.260806 | 0.332760 | 0.2601866 | 0.230814 | 0.295530 | 0.2381250 |

160 | 0.498065 | 0.492069 | 0.4886004 | 0.266994 | 0.325688 | 0.2682712 | 0.233976 | 0.286045 | 0.2438489 |

180 | 0.498793 | 0.507865 | 0.4879055 | 0.278340 | 0.339096 | 0.2791945 | 0.242351 | 0.298429 | 0.2519019 |

200 | 0.491820 | 0.505140 | 0.4673425 | 0.289293 | 0.341454 | 0.2788359 | 0.252402 | 0.297941 | 0.2501888 |

Window Size | Precision | Recall | F1 Score | ||||||
---|---|---|---|---|---|---|---|---|---|

DILOF | TADILOF | MILOF | DILOF | TADILOF | MILOF | DILOF | TADILOF | MILOF | |

100 | 0.00265890 | 0.00164838 | 0.002386766 | 0.7633 | 0.7400 | 0.5029 | 0.00525291 | 0.00327270 | 0.004681796 |

200 | 0.00256851 | 0.00247500 | 0.002520710 | 0.7733 | 0.7900 | 0.5147 | 0.00507621 | 0.00491061 | 0.004933168 |

300 | 0.00332737 | 0.00344192 | 0.002814311 | 0.7933 | 0.8133 | 0.6179 | 0.00655182 | 0.00679199 | 0.005521916 |

400 | 0.00265894 | 0.00238982 | 0.002669190 | 0.7867 | 0.9133 | 0.6603 | 0.00525692 | 0.00475252 | 0.005260103 |

500 | 0.00191050 | 0.00313257 | 0.002218161 | 0.7467 | 0.9467 | 0.6417 | 0.00379399 | 0.00620751 | 0.004383276 |

600 | 0.00203202 | 0.00191777 | 0.001829777 | 0.7700 | 0.9767 | 0.5839 | 0.00403406 | 0.00382331 | 0.003621248 |

700 | 0.00168554 | 0.00185686 | 0.001824417 | 0.6767 | 0.9067 | 0.5933 | 0.00334839 | 0.00369957 | 0.003614118 |

Window Size | Precision | Recall | F1 Score | ||||||
---|---|---|---|---|---|---|---|---|---|

DILOF | TADILOF | MILOF | DILOF | TADILOF | MILOF | DILOF | TADILOF | MILOF | |

100 | 0.14336408 | 0.1570996 | 0.1922093 | 0.3256 | 0.3898 | 0.4302 | 0.1121199 | 0.130352 | 0.179416 |

120 | 0.16889650 | 0.1551854 | 0.1959132 | 0.3476 | 0.4350 | 0.4202 | 0.1239128 | 0.148712 | 0.171690 |

140 | 0.16830130 | 0.1644227 | 0.2006647 | 0.3660 | 0.4604 | 0.4350 | 0.1329999 | 0.158371 | 0.175122 |

160 | 0.17210987 | 0.1958837 | 0.2394359 | 0.3756 | 0.4758 | 0.4384 | 0.1360716 | 0.166075 | 0.179563 |

180 | 0.16043631 | 0.1741156 | 0.2275521 | 0.3862 | 0.5022 | 0.4494 | 0.1367308 | 0.174471 | 0.182075 |

200 | 0.16436960 | 0.1809000 | 0.2074316 | 0.3914 | 0.5000 | 0.4348 | 0.1390244 | 0.173421 | 0.173099 |

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## Share and Cite

**MDPI and ACS Style**

Huang, J.-W.; Zhong, M.-X.; Jaysawal, B.P. TADILOF: Time Aware Density-Based Incremental Local Outlier Detection in Data Streams. *Sensors* **2020**, *20*, 5829.
https://doi.org/10.3390/s20205829

**AMA Style**

Huang J-W, Zhong M-X, Jaysawal BP. TADILOF: Time Aware Density-Based Incremental Local Outlier Detection in Data Streams. *Sensors*. 2020; 20(20):5829.
https://doi.org/10.3390/s20205829

**Chicago/Turabian Style**

Huang, Jen-Wei, Meng-Xun Zhong, and Bijay Prasad Jaysawal. 2020. "TADILOF: Time Aware Density-Based Incremental Local Outlier Detection in Data Streams" *Sensors* 20, no. 20: 5829.
https://doi.org/10.3390/s20205829