# Assessing Feature Representations for Instance-Based Cross-Domain Anomaly Detection in Cloud Services Univariate Time Series Data

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## Abstract

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## 1. Introduction

- A comprehensive analysis of the multiple feature time-series feature sets for anomaly detection across a number of cross-domain time-series datasets;
- Identifying a feature set that outperforms the other representations by a statistically significant margin on the task of cross-domain anomaly detection;
- Elaborating feature importance in the top performing representation.

## 2. Related Work on Feature Engineering for Time-Series Analysis and Anomaly Detection

## 3. Feature Sets under Test

- A list of candidate periods are extracted from the time series using the periodogram method, and
- The candidate periods are then ranked using an autocorrelation method ACF.

- Spectral Residual (SR) is the concept borrowed from the visual computing world which is based on Fast Fourier Transform (FFT). Spectral residual is unsupervised and has proven its efficacy in visual saliency detection applications. In anomaly detection tasks, the anomalies in the time series have similar characteristics to saliency detection, i.e., they both visually stand out from the normal distribution.
- Skewness is the measurement of the symmetry in a frequency distribution. Skewness is very sensitive to outlier values because the calculation of skewness depends on the cubed distance between a value and the mean. Hence, we believe that the presence of any outlier may affect the skewness of a distribution within a time window, and so skewness might be helpful in anomaly detection tasks. Indeed, the authors of [36] propose an approach to anomaly detection based on skewness.
- Kurtosis is often described as the extent to which the peak of a probability distribution deviates from the shape of a normal distribution, i.e., whether the shape is too peaked or too flat to be normally distributed. This feature can be used as a complementary feature with the skewness in case of distributions that are symmetrical but are too peaked or too flat to be normally distributed. In fact, it has been mathematically proven in the context of dimensionality reduction techniques for anomaly detection that the optimal directions of projection of a time series into a reduced dimensionality are those that maximize or minimize the kurtosis coefficient of the project time series (see [37,38]), and [39] is an example of recent work that proposes an approach to anomaly detection in time-series data based on kurtosis.

- Raw (window size = 15),
- HAD
- catch22,
- catch23 (catch22 + Spectral Residual),
- catch24 (catch22 + Mean + Variance),
- catch25 (catch22 + SR + Mean + Variance).

## 4. Experimental Design

- Naïve bayes: Model parameters are inferred from training data.
- Random forest: 200 estimators, $max\_features=\sqrt{n\_features}$, no max depth, two samples as minimum number of samples required to split an internal node, one as minimum number of samples in newly created leaves, with bootstrapping, using out-of-bag samples to estimate the generalization error.
- Xgboost: gbtree booster, step size shrinkage of $0.3$, 0 Minimum loss reduction required to make a further partition on a leaf node of the tree, 6 maximum depth of a tree,
- Multi-layer perceptron: 1 hidden layer with 100 ReLU units per layer with a single logistic unit in the output layer, adam as the solver, using L2 penalty of 0.0001, batch size of 200, constant learning rate, 0.001 step-size in updating the weights, max iterations set to 200, no early stopping, 0.9 as exponential decay rate for estimates of first moment vector, and 0.999 as exponential decay rate for estimates of second moment vector, 1 × 10${}^{-8}$ for numerical stability.
- SVC: rbf kernel, regularization parameter of 1.0, scaled kernel coefficient on data set variance, and vector length.
- SVC linear: linear kernel, and regularization parameter of 0.25.
- -
- Other SVC parameters: shrinking heuristic, no probability estimates, 1 × 10${}^{-3}$ tolerance for stopping criterion, 200 MB kernel cache, no class weights, no limit on number of iterations, and one-vs.-rest decision function.

- NAB (AWS and Twitter) is a benchmark for evaluating anomaly detection algorithms in streaming, real-time applications. It is composed of over 50 labeled real-world and artificial time series. We focus on two of the real-world datasets: (i) web monitoring statistics from AWS and (ii) Twitter tweet volumes. The AWS time series are taken from different server metrics such as CPU utilization, network traffic, and disk write bytes. The Twitter dataset is the number of Twitter mentions of publicly-traded companies such as Google and IBM. The value represents the number of mentions for a given ticker symbol in every five windows.
- Yahoo Anomaly Detection Dataset is a publicly available collection of datasets released by Yahoo for benchmarking anomaly detection algorithms. The dataset has four sets of time series: one is collected from production traffic to yahoo services, which is labelled by editors manually, whereas the others are synthesized with anomalies that are embedded artificially. We use the real data and synthesised data but not the “A3” and “A4” datasets.
- IOPS KPI dataset is a collection of time series datasets provided by Alibaba, Tencent, Baidu, eBay, and Sogou. The dataset is from real traffic on web services and was published as part of a series of anomaly detection competitions.
- Huawei dataset is the dataset obtained from the anomaly detection hackathon organised by Huawei (Details of the Hackathon competition are available at: https://huawei-euchallenge.bemyapp.com/ireland (accessed on 2 December 2020)). The selected time series contains different KPI values, and each datapoint is labelled as either anomaly or not anomaly.

Algorithm 1: Structure of Experimental Design |

## 5. Results

#### Feature Importance by Random Forest

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Table A1.**The catch22 feature set (based on Table 1 from [30] and reproduced here with minor formatting and wording modifications under the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/)).

Feature | Description |
---|---|

DN HistogramMode 5 | Mode of z-scored distribution (5-bin histogram) |

DN HistogramMode 10 | Mode of z-scored distribution (10-bin histogram) |

SB BinaryStats mean longstretch1 | Longest period of consecutive values above the mean |

DN OutlierInclude p 001 mdrmd | Time intervals between successive extreme event above the mean |

DN OutlierInclude n 001 mdrmd | Time intervals between successive extreme events below the mean |

CO f1ecac | First 1/e crossing of autocorrelation function |

CO FirstMin ac | First minimum of autocorrelation function |

SP Summaries welch rect area 5 1 | Total power in lowest fifth of frequencies in the Fourier power spectrum |

SP Summaries welch rect centroid | Centroid of the Fourier power spectrum |

FC LocalSimple mean3 stderr | Mean error from a rolling 3-sample mean |

IN AutoMutualInfoStats 40 gaussian fmmi | First minimum of the automutual information function |

CO HistogramAMI even 2 5 | Automutual information, $m=2,\tau =5$ |

CO trev 1 num | Time-reversibility statistic, ${\u2329{\left({x}_{t+1}-{x}_{t}\right)}^{3}\u232a}_{t}$ |

MD hrv classic pnn40 | Proportion of successive differences exceeding $0.04\sigma $ |

SB BinaryStats diff longstretch0 | Longest period of successive incremental decreases |

SB MotifThree quantile hh | Shannon entropy of two successive letters in equiprobable 3 letter symbolization |

FC LocalSimple mean1 tauresrat | Change in correlation length after iterative differencing |

CO Embed2 Dist tau d expfit meandiff | Exponential fit to successive distances in 2-d embedding space |

SC FluctAnal 2 dfa 50 1 2 logi prop r1 | Proportion of slower time scale fluctuations that scale with DFA ($50\%$ sampling) |

SC FluctAnal 2 rsrangefit 50 1 logi prop r1 | Proportion of slower time scale fluctuations that scale with linearly rescaled range fit |

5SB TransitionMatrix 3ac sumdiagcov | Trace of covariance of transition matrix between symbols in 3-letter alphabet |

PD PeriodicityWang th0 01 | Periodicity |

**Table A2.**The statistical feature set (based on Table 1 in [4]).

Feature | Description |
---|---|

ACF1 | First order of autocorrelation |

ACF Remainder | Autocorrelation of remainder |

Mean | Rolling mean |

Variance | Rolling variance |

Entropy | Spectral entropy |

Linearity | Strength of linearity |

Trend | Strength trend |

Crossing Point | number of crossing points |

ARCHtest | p value of Lagrange Multiplier (LM) test for ARCH model |

Curvature | Strength of curvature computed on Trend of STL decomposition |

GARCHtest.p | p value of Lagrange Multiplier |

**Table A3.**Forecasting error features (based on Table 2 in [4]).

Feature | Formula | Description |
---|---|---|

RMSE | $\sqrt{\frac{\sum {\left({Y}_{i}-{\widehat{Y}}_{i}\right)}^{2}}{N}}$ | Root mean square error |

ME | $\frac{\sum \left({Y}_{i}-{\widehat{Y}}_{i}\right)}{N}$ | Mean error |

MAE | $\frac{\sum |{Y}_{i}-{\widehat{Y}}_{i}|}{N}$ | Mean absolute error |

MPE | $\frac{1}{N}\xb7{\sum}_{i=1}^{N}\frac{{Y}_{i}-{\widehat{Y}}_{i}}{{Y}_{i}}$ | Mean percentage error |

MAPE | $\frac{1}{N}\xb7{\sum}_{i=1}^{N}\frac{|{Y}_{i}-{\widehat{Y}}_{i}|}{{Y}_{i}}$ | Mean absolute percentage error |

**Table A4.**Temporal features set (based on Table 3 in [4]).

Feature | Description |
---|---|

Max level shift | Max trimmed mean between two consecutive windows |

Max var shift | Max variance shift between two consecutive windows |

Max KL shift | Max shift in Kullback–Leibler divergence between two consecutive windows |

Diff-w | The differences between the current value and the w-th previous value |

Lumpiness | Changing variance in remainder |

Flatspots | Discretize time series value into ten equal sized interval, find maximum run length within the same bucket |

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**Figure 1.**The combinations of datasets, feature sets, and models used in the experiments. Non-standard terminology and acronyms: catch23 is catch22+sr, catch24 is catch22+mean+variance, catch25 is catch22+sr+mean+variance, random forest (RF), Naïve Bayes (NB), multi-layer perceptron (MLP), radial basis function (RBF), and multilayer perceptron (MLP).

**Table 1.**The HAD feature set, created by adapting the feature set of [4]. Notation: N number of data points in a time series and/or set of predictions, ${\mu}_{T}$ mean of values in time series T, ${\sigma}_{T}$ standard deviation of values in time series T, ${Y}_{i}$ the observed value at time i, ${\widehat{Y}}_{i}$ the estimated value for time i

Feature | Description |
---|---|

ACF1 | First order of autocorrelation |

ACF Remainder | Autocorrelation of remainder |

Mean | Mean of the rolling time series window |

Variance | Variance of the rolling time series window |

Entropy | Spectral entropy (see [16]) |

Linearity | Strength of linearity computed on the trend of STL decomposition (see [16]) |

Trend | Strength of trend |

Crossing Point | Number of crossing points (see [16]) |

ARCHtest.p | P value of Lagrange Multiplier (LM) test for ARCH model (see [20]) |

Curvature | Strength of curvature computed on Trend of STL decomposition (from [16]) |

RMSE | Root mean square error $\sqrt{\frac{{\sum}_{i=1}^{N}{\left({Y}_{i}-{\widehat{Y}}_{i}\right)}^{2}}{N}}$ |

ME | Mean error $\frac{{\sum}_{i=1}^{N}\left({Y}_{i}-{\widehat{Y}}_{i}\right)}{N}$ |

MAE | Mean absolute error $\frac{{\sum}_{i=1}^{N}|{Y}_{i}-{\widehat{Y}}_{i}|}{N}$ |

MPE | Mean percentage error $\frac{1}{N}\xb7{\sum}_{i=1}^{N}\frac{{Y}_{i}-{\widehat{Y}}_{i}}{{Y}_{i}}$ |

MAPE | Mean absolute percentage error $\frac{1}{N}\xb7{\sum}_{i=1}^{N}\frac{|{Y}_{i}-{\widehat{Y}}_{i}|}{{Y}_{i}}$ |

Max level shift | Max trimmed mean between two consecutive windows |

Max var shift | Max variance shift between two consecutive windows |

Max KL shift | Max shift in Kullback–Leibler divergence between two consecutive windows |

Diff-w | The differences between the current value and the w-th previous value |

SR | For time point i the value calculated from the SR transformed time series: |

$S{R}_{i}=\sqrt{S{R}_{i}.rea{l}^{2}+S{R}_{i}.imaginar{y}^{2}}$ | |

Skewness | Skewness coefficient of the rolling time series window $T={Y}_{1},\cdots ,{Y}_{N}$: ${S}_{T}=\frac{{\sum}_{i=1}^{N}{({Y}_{i}-{\mu}_{T})}^{3}/N}{{\sigma}_{T}^{3}}$ |

Kurtosis | Kurtosis coefficient of the rolling time series window $T={Y}_{1},\cdots ,{Y}_{N}$: ${K}_{T}=\frac{{\sum}_{i=1}^{N}{({Y}_{i}-{\mu}_{T})}^{4}/N}{{\sigma}_{T}^{4}}$ |

Dataset | Number of Points | % Anomalies | Number of Time Series | Mean Length |
---|---|---|---|---|

Yahoo Real | 91 K | 1.76% | 64 | 1415 |

Yahoo Art | 140 K | 1.76% | 100 | 1415 |

IOPS | 3 M | 2.26% | 29 | 105,985 |

AWS | 67 K | 4.57% | 17 | 67,740 |

142 K | 0.15% | 10 | 142,765 | |

Huawei | 54 K | 4.19% | 6 | 9056 |

**Table 3.**For each feature set under test, the mean F1, recall, and precision, along with confidence intervals calculated to 95% and range calculated across model types and dataset fold combinations (populations sizes: 157, 179, 172, 172, 151, and 151). Bold font indicates highest score for a metric.

F1 | Recall | Precision | |||||||
---|---|---|---|---|---|---|---|---|---|

Avg | Conf | Range | Avg | Conf | Range | Avg | Conf | Range | |

Raw (ws = 15) | 0.2558 | ±0.0390 | 0.2168 0.2948 | 0.2263 | ±0.0389 | 0.1873 0.2652 | 0.5604 | ±0.0565 | 0.5039 0.6169 |

HAD | 0.6481 | ±0.0464 | 0.6016 0.6945 | 0.6297 | ±0.0465 | 0.5833 0.6762 | 0.7818 | ±0.0460 | 0.7358 0.8278 |

catch22 | 0.5537 | ±0.0528 | 0.5009 0.6066 | 0.5132 | ±0.0516 | 0.4617 0.5648 | 0.6993 | ±0.0541 | 0.6452 0.7534 |

catch23 | 0.5506 | ±0.0525 | 0.4980 0.6031 | 0.509 | ±0.0511 | 0.4579 0.5601 | 0.699 | ±0.0542 | 0.6447 0.7532 |

catch24 | 0.5889 | ±0.0614 | 0.5275 0.6503 | 0.5407 | ±0.0608 | 0.4799 0.6015 | 0.7826 | ±0.0507 | 0.7319 0.8333 |

catch25 | 0.5825 | ±0.0612 | 0.5213 0.6436 | 0.532 | ±0.0608 | 0.4712 0.5928 | 0.7812 | ±0.0504 | 0.7308 0.8316 |

**Table 4.**Mean F1 by dataset, confidence interval calculated to 95% with population sizes 165, 180, 137, 165, 170, and 165, respectively. Bold font indicates highest average F1 score.

Average | Confidence | ||
---|---|---|---|

Data | F1 | (Error Margin) | Range |

AWS | 0.3963 | ±0.0549 | 0.3414 0.4512 |

Huawei | 0.5830 | ±0.0553 | 0.5278 0.6383 |

IOPS | 0.5693 | ±0.0513 | 0.5180 0.6206 |

0.3729 | ±0.0476 | 0.3253 0.4205 | |

Yahoo Artificial | 0.7724 | ±0.0392 | 0.7333 0.8116 |

Yahoo Real | 0.4958 | ±0.0593 | 0.4365 0.5551 |

**Table 5.**Mean F1 by model type, confidence interval calculated to 95% with population sizes 180, 180, 180, 163, 99, and 180, respectively. Bold font indicates highest average F1 score.

Average | Confidence | ||
---|---|---|---|

Data | F1 | (Error Margin) | Range |

Multi-Layer Perceptron | 0.6129 | ±0.0451 | 0.5677 0.6580 |

Naïve Bayes | 0.2386 | ±0.0242 | 0.2144 0.2627 |

Random Forest | 0.8279 | ±0.03110 | 0.7967 0.8590 |

SVC | 0.3259 | ±0.04680 | 0.2791 0.3727 |

SVC Linear | 0.2233 | ±0.0710 | 0.1524 0.2943 |

Xgboost | 0.8082 | ±0.0322 | 0.7760 0.8404 |

**Table 6.**For each feature set under test, the mean F1, recall, and precision, along with confidence intervals, were used to calculate the 95% interval (in all cases using a population of 50 samples), and range was calculated across the results for the Random Forest and Xgboost model types and dataset fold (excluding Yahoo Artificial) combinations. Bold font indicates highest score for a metric.

F1 | Recall | Precision | |||||||
---|---|---|---|---|---|---|---|---|---|

Avg | Conf | Range | Avg | Conf | Range | Avg | Conf | Range | |

Raw (ws = 15) | 0.4253 | ±0.0700 | 0.3553 0.4954 | 0.3236 | ±0.0693 | 0.2543 0.3930 | 0.7800 | ±0.0445 | 0.7355 0.8245 |

HAD | 0.8675 | ±0.0269 | 0.8406 0.8944 | 0.8030 | ±0.0378 | 0.7652 0.8408 | 0.9556 | ±0.0144 | 0.9412 0.9700 |

catch22 | 0.8434 | ±0.0315 | 0.8119 0.8749 | 0.7610 | ±0.0419 | 0.7191 0.8029 | 0.9598 | ±0.0120 | 0.9478 0.9718 |

catch23 | 0.8360 | ±0.0313 | 0.8047 0.8673 | 0.7495 | ±0.0413 | 0.7082 0.7909 | 0.9587 | ±0.0123 | 0.9464 0.9710 |

catch24 | 0.9167 | ±0.0196 | 0.8971 0.9364 | 0.8678 | ±0.0283 | 0.8395 0.8960 | 0.9766 | ±0.0070 | 0.9696 0.9835 |

catch25 | 0.9161 | ±0.0198 | 0.8964 0.9359 | 0.8671 | ±0.0284 | 0.8387 0.8954 | 0.9760 | ±0.0073 | 0.9687 0.9832 |

**Table 7.**Mean F1 Recall and Precision of the Random Forest model on the Huawei and IOPS datsets using the catch25 and HAD feature sets, respectively. Bold font indicates highest F1 on a dataset.

Huawei | IOPS | |||||
---|---|---|---|---|---|---|

Prec. | Recall | F1 | Prec. | Recall | F1 | |

catch25 | 0.9940 | 0.9465 | 0.9697 | 0.9908 | 0.8957 | 0.9409 |

HAD | 0.9881 | 0.8759 | 0.9287 | 0.9897 | 0.9069 | 0.9465 |

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## Share and Cite

**MDPI and ACS Style**

Agrahari, R.; Nicholson, M.; Conran, C.; Assem, H.; Kelleher, J.D. Assessing Feature Representations for Instance-Based Cross-Domain Anomaly Detection in Cloud Services Univariate Time Series Data. *IoT* **2022**, *3*, 123-144.
https://doi.org/10.3390/iot3010008

**AMA Style**

Agrahari R, Nicholson M, Conran C, Assem H, Kelleher JD. Assessing Feature Representations for Instance-Based Cross-Domain Anomaly Detection in Cloud Services Univariate Time Series Data. *IoT*. 2022; 3(1):123-144.
https://doi.org/10.3390/iot3010008

**Chicago/Turabian Style**

Agrahari, Rahul, Matthew Nicholson, Clare Conran, Haytham Assem, and John D. Kelleher. 2022. "Assessing Feature Representations for Instance-Based Cross-Domain Anomaly Detection in Cloud Services Univariate Time Series Data" *IoT* 3, no. 1: 123-144.
https://doi.org/10.3390/iot3010008