Hybrid Forecasting Anomaly Detection and Interbank ...

Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

Hybrid Forecasting Anomaly Detection and Interbank Interest Rates Carlo Drago* and Paolo D’Ambra**

*University of Rome “Niccolo Cusano”, **Independent Scholar

V International Workshop on Computational Economics and Econometrics CNR Rome 19 June 2017

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

Outline

Research problem

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

Outline

Research problem Methodology

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

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Research problem Methodology Simulation study

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

Outline

Research problem Methodology Simulation study Conclusions and directions for future research

Outline

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

Outline

Research problem Methodology Simulation study Conclusions and directions for future research

Outline

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

The Research Problem

Anomaly Detection in The Financial Risk Management

Anomaly detection is a family of different approaches related to find the deviations from a given pattern on the data or an specified expected behaviour. (Chandola et al. 2009). Anomaly detection can be used to identify anomalies in financial data that could be attributed to fraud or manipulation. It can be a useful tool to manage risk and eventually anticipate it. Interbank interest rates and hybrid forecasting We focused on anomalies among interbank interest rate time series proposing an hybrid forecasting approach using an automatic-ARIMA algorithm (Hyndman 2017 Hyndman Khandakar 2008) and an autoregressive neural network (Zhang 2009 De Nadai Someren 2015).

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

The Research Problem

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Anomaly Detection in The Financial Risk Management

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

The Research Problem

Interest Rates and Anomaly Detection

Between 2005 and 2008 interbank interest rate increased sharply causing high interest rate to be paid for loans by householders; meanwhile, institutionals and investment banks were purcheasing financial derivatives to hedge or speculate on floating interest rate. The 16 of April (2008) a Wall Street Journal article came up warning about a possible Libor manipulation. (see Hou, David; Skeie, David 2014) Most traded financial asset is indexed at interbank interest rate and their market share, for instance Libor, in 2013 has been estimated to be indexed in more than 300 trillion of dollar among financial contracts, loans and mortgage (see Hou, David, Skeie, David 2014)

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

The Research Problem

A brief Introduction to Interbank Interest Rate

The interbank interest rate is a floating rate and figures as the benchmark and the average rate of all rates comunicated by prime banks of the panel. Prime banks refers to this benchmark to lend unsecured funds to other prime banks, with a certain maturity, who needs to manage liquidity and facing at reserve requirements (see Gaspar, Quiros and Mendizabal 2004). The average rate reflects the credit rating and liquidity position of all panel banks involved in its estimation Funding operations among banks occur in the interbank market

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

A brief Introduction to interbank interest rate

The Research Problem

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

The Research Problem

The official value of the interbank interest rate: EURIBOR

The process: Each bank belonging to the panel communicate an estimation of the Euribor according to the expectations that each bank has for the deposit operation. The data is transferred to the Trans European Automated Realtime Gross Settlement Express Transfer System and than is managed by the staff of Thomson Reuters in order to fix the data comunicated. The same data, before being officially communicated, are treated by a fixing process who provides to eliminate the 15 % of all the topping and tailing values of the acquired data. When these steps are completed, the Thomson Reuters team continue the fixing process through the average estimation of the treated data. The interbank rate so calculated will be officially communicate to the financial comunity.

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

The raise of interbank interest rate and its aftermath

The Research Problem

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

The Research Problem

Troubles and effects of the raise: focus on italian households

Households loans background: Loans issued reached the amount of 220-230 billions of euro Interest quote commisured to euribor was 30 billions of euro Borrowers damage is estimated to be over 3 billion euro to 2.5 million Italian families with an average of 1.200 euro per capita damage

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

Anomaly Detection

Anomaly Detection Recent technological developments have made possible collect a very large amount of data. At the same time there is a relevant need of the different organization to detect data anomalies on this data flow (Hyndman et al. 2015). The time series decomposition The decomposition of the time series and their separate treatment it is very useful in order to identifying also the different sources of anomaly which can occur.

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

Anomaly Detection

Anomaly Detection Recent technological developments have made possible collect a very large amount of data. At the same time there is a relevant need of the different organization to detect data anomalies on this data flow (Hyndman et al. 2015). The time series decomposition The decomposition of the time series and their separate treatment it is very useful in order to identifying also the different sources of anomaly which can occur. Research on this topic seems to be related to other important topics in time series analysis: novelty detection, identification of structural changes, control charts and so on (Garziano 2016). In particular the problem of data cleaning seems to be a very relevant issue (Rahm and Do 2000).

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

Anomaly Detection

Anomaly Detection Recent technological developments have made possible collect a very large amount of data. At the same time there is a relevant need of the different organization to detect data anomalies on this data flow (Hyndman et al. 2015). The time series decomposition The decomposition of the time series and their separate treatment it is very useful in order to identifying also the different sources of anomaly which can occur. Research on this topic seems to be related to other important topics in time series analysis: novelty detection, identification of structural changes, control charts and so on (Garziano 2016). In particular the problem of data cleaning seems to be a very relevant issue (Rahm and Do 2000). At the same time it is important to detect in real time relevant changes or anomalies on the processes.

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

Anomaly Detection

Anomaly Detection Recent technological developments have made possible collect a very large amount of data. At the same time there is a relevant need of the different organization to detect data anomalies on this data flow (Hyndman et al. 2015). The time series decomposition The decomposition of the time series and their separate treatment it is very useful in order to identifying also the different sources of anomaly which can occur. Research on this topic seems to be related to other important topics in time series analysis: novelty detection, identification of structural changes, control charts and so on (Garziano 2016). In particular the problem of data cleaning seems to be a very relevant issue (Rahm and Do 2000). At the same time it is important to detect in real time relevant changes or anomalies on the processes. ...in this context arise the anomaly detection approaches (Chandola et al. 2009, Cheboli 2010 between other contributions):

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

Anomaly Detection

Anomaly Detection Anomalies Anomalies can be considered observations or group of observations which are not specifically related a specific pattern on data They should be not considered necessarily outliers. In particular outliers are single observations in these cases we can observe ”strange” subsequences of data. They are difficult to detect on a specific time series.

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

Anomaly Detection

Anomaly Detection Anomalies Anomalies can be considered observations or group of observations which are not specifically related a specific pattern on data They should be not considered necessarily outliers. In particular outliers are single observations in these cases we can observe ”strange” subsequences of data. They are difficult to detect on a specific time series. In these cases an approach specific on these cases it is necessary.

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

Anomaly Detection

Anomaly Detection Anomalies Anomalies can be considered observations or group of observations which are not specifically related a specific pattern on data They should be not considered necessarily outliers. In particular outliers are single observations in these cases we can observe ”strange” subsequences of data. They are difficult to detect on a specific time series. In these cases an approach specific on these cases it is necessary. In this sense: Anomaly Detection The anomaly detection approaches are useful to identify statistical units defined as anomalies on a definite pattern in a dataset. Clearly it is necessary to identify a normal behaviour before.

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

Anomaly Detection

Anomaly Detection Anomalies Anomalies can be considered observations or group of observations which are not specifically related a specific pattern on data They should be not considered necessarily outliers. In particular outliers are single observations in these cases we can observe ”strange” subsequences of data. They are difficult to detect on a specific time series. In these cases an approach specific on these cases it is necessary. In this sense: Anomaly Detection The anomaly detection approaches are useful to identify statistical units defined as anomalies on a definite pattern in a dataset. Clearly it is necessary to identify a normal behaviour before. Various different techniques were proposed in literature in order to perform anomaly detection on various frameworks and application fields.

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

Anomaly Detection

Anomaly Detection

In this work we consider anomalies on time series data. It is possible consider anomaly detection also in other contexts. The relevance of the anomaly detection here cannot be undervalued and there can be considered in many different applications on real contexts. For example applications can be found: biosurveillance health

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

Anomaly Detection

Anomaly Detection

In this work we consider anomalies on time series data. It is possible consider anomaly detection also in other contexts. The relevance of the anomaly detection here cannot be undervalued and there can be considered in many different applications on real contexts. For example applications can be found: biosurveillance health network monitoring intrusion Lotze 2009 Vishwasrao et al. (2016) and also cloud data (Vallis, Hochenbaum and Kejariwa 2014)

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

Anomaly Detection

Anomaly Detection

In this work we consider anomalies on time series data. It is possible consider anomaly detection also in other contexts. The relevance of the anomaly detection here cannot be undervalued and there can be considered in many different applications on real contexts. For example applications can be found: biosurveillance health network monitoring intrusion Lotze 2009 Vishwasrao et al. (2016) and also cloud data (Vallis, Hochenbaum and Kejariwa 2014) The topic is also connected with outlier identification on a time series (Hodge and Austin 2004) which have at the same time many relevant applications.

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

Anomaly Detection

Anomaly Detection on Time Series

Various different anomalies can be detected on time series (Cheboli 2010, Chandola et al. 2009 and Lotze 2009):

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

Anomaly Detection

Anomaly Detection on Time Series

Various different anomalies can be detected on time series (Cheboli 2010, Chandola et al. 2009 and Lotze 2009): Contextual anomalies on the time series

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

Anomaly Detection

Anomaly Detection on Time Series

Various different anomalies can be detected on time series (Cheboli 2010, Chandola et al. 2009 and Lotze 2009): Contextual anomalies on the time series Anomalous subsequences of time series

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

Anomaly Detection

Anomaly Detection on Time Series

Various different anomalies can be detected on time series (Cheboli 2010, Chandola et al. 2009 and Lotze 2009): Contextual anomalies on the time series Anomalous subsequences of time series Anomalies related a group of time series

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

Anomaly Detection

Anomaly Detection on Time Series

Various different anomalies can be detected on time series (Cheboli 2010, Chandola et al. 2009 and Lotze 2009): Contextual anomalies on the time series Anomalous subsequences of time series Anomalies related a group of time series In all these cases we need to consider different approach and strategies to identify the anomalies

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

Anomaly Detection

Anomaly Detection on Time Series

There are many different approaches, algorithms and methodologies to perform anomaly detection on time series. Chandola et al. 2009, Cheboli (2010) propose a survey of the different methods which are used on time series. In particular different approaches can be related to approaches based on: Windows Based Techniques

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

Anomaly Detection

Anomaly Detection on Time Series

There are many different approaches, algorithms and methodologies to perform anomaly detection on time series. Chandola et al. 2009, Cheboli (2010) propose a survey of the different methods which are used on time series. In particular different approaches can be related to approaches based on: Windows Based Techniques Proximity or Kernel Based Techniques

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

Anomaly Detection

Anomaly Detection on Time Series

There are many different approaches, algorithms and methodologies to perform anomaly detection on time series. Chandola et al. 2009, Cheboli (2010) propose a survey of the different methods which are used on time series. In particular different approaches can be related to approaches based on: Windows Based Techniques Proximity or Kernel Based Techniques Prediction Techniques

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

Anomaly Detection

Anomaly Detection on Time Series

There are many different approaches, algorithms and methodologies to perform anomaly detection on time series. Chandola et al. 2009, Cheboli (2010) propose a survey of the different methods which are used on time series. In particular different approaches can be related to approaches based on: Windows Based Techniques Proximity or Kernel Based Techniques Prediction Techniques Hidden Markov Models

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

Anomaly Detection

Anomaly Detection on Time Series

There are many different approaches, algorithms and methodologies to perform anomaly detection on time series. Chandola et al. 2009, Cheboli (2010) propose a survey of the different methods which are used on time series. In particular different approaches can be related to approaches based on: Windows Based Techniques Proximity or Kernel Based Techniques Prediction Techniques Hidden Markov Models Segmentation Based Techniques

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

Anomaly Detection

Anomaly Detection on Time Series

There are many different approaches, algorithms and methodologies to perform anomaly detection on time series. Chandola et al. 2009, Cheboli (2010) propose a survey of the different methods which are used on time series. In particular different approaches can be related to approaches based on: Windows Based Techniques Proximity or Kernel Based Techniques Prediction Techniques Hidden Markov Models Segmentation Based Techniques Feature Based Techniques

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

Anomaly Detection

Anomaly Detection on Time Series

There are many different approaches, algorithms and methodologies to perform anomaly detection on time series. Chandola et al. 2009, Cheboli (2010) propose a survey of the different methods which are used on time series. In particular different approaches can be related to approaches based on: Windows Based Techniques Proximity or Kernel Based Techniques Prediction Techniques Hidden Markov Models Segmentation Based Techniques Feature Based Techniques Others related different methodologies About feature based techniques category see Vallis (2014) and Goldberg, Shan (2015) about other approaches see Garziano (2016)

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

Anomaly Detection

Methodology

In this sense we consider a strategy of anomaly detection for time series data. The algorithm can be described as follow: Phase I: identification of the statistical pattern Hybrid Forecasting of the time series considered using an Auto-Arima approach (component A). See Hyndman 2017 and Hyndman Khandakar 2008.

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

Anomaly Detection

Methodology

In this sense we consider a strategy of anomaly detection for time series data. The algorithm can be described as follow: Phase I: identification of the statistical pattern Hybrid Forecasting of the time series considered using an Auto-Arima approach (component A). See Hyndman 2017 and Hyndman Khandakar 2008. then we consider an Autoregressive Neural Network in order to analyze the residuals of the series (component B)

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

Anomaly Detection

Methodology

In this sense we consider a strategy of anomaly detection for time series data. The algorithm can be described as follow: Phase I: identification of the statistical pattern Hybrid Forecasting of the time series considered using an Auto-Arima approach (component A). See Hyndman 2017 and Hyndman Khandakar 2008. then we consider an Autoregressive Neural Network in order to analyze the residuals of the series (component B) Finally we obtain the estimation of the original time series from the components A and B (Zhang 2003) Phase II: identification of the anomalies From the analysis of the identified patterns we can detect the relevant anomalies over time.

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

Anomaly Detection

Methodology

In this sense we consider a strategy of anomaly detection for time series data. The algorithm can be described as follow: Phase I: identification of the statistical pattern Hybrid Forecasting of the time series considered using an Auto-Arima approach (component A). See Hyndman 2017 and Hyndman Khandakar 2008. then we consider an Autoregressive Neural Network in order to analyze the residuals of the series (component B) Finally we obtain the estimation of the original time series from the components A and B (Zhang 2003) Phase II: identification of the anomalies From the analysis of the identified patterns we can detect the relevant anomalies over time. Sensitivity analysis of the obtained results

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

Anomaly Detection

Methodology

In this sense we consider a strategy of anomaly detection for time series data. The algorithm can be described as follow: Phase I: identification of the statistical pattern Hybrid Forecasting of the time series considered using an Auto-Arima approach (component A). See Hyndman 2017 and Hyndman Khandakar 2008. then we consider an Autoregressive Neural Network in order to analyze the residuals of the series (component B) Finally we obtain the estimation of the original time series from the components A and B (Zhang 2003) Phase II: identification of the anomalies From the analysis of the identified patterns we can detect the relevant anomalies over time. Sensitivity analysis of the obtained results

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

Anomaly Detection

The Methodology

Identification of the statistical pattern: hybrid forecasting Following Zhang (2003), we estimate the following ARIMA model yt = γ0 − α1 xt−1 − · · · − αp xt−p + εt − θ1 εt−1 + · · · − θq εt−q

(1)

Then, from the auto-ARIMA model we can assume we are able to represent the linear structure of the time series. In this sense the time series can be decomposed on two parts: yt = Lt + Nt

(2)

et = yt − Lˆt

(3)

The residuals are:

From the following expression which represent the way we can to decompose the original interest rates. yˆt = Lˆt + Nˆt

(4)

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

Anomaly Detection

The Methodology

Anomaly detection We start to considering the time series we model the series by considering an auto-arima approach then we model the residuals with an autoregressive neural network. Finally we add the two estimated components. In order to approach to hybrid forecasting anomaly detection we consider: De Nadai Someren (2015). In this sense we identify the anomalies by subtracting the obtained prediction from the original time series. The observations which are showing values highly different are considered anomalies. So we have: At = yt − yˆt

(5)

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

Anomaly Detection

Simulation study

The structure of the simulation We simulate some realistic cases with the aim to show the characteristics of the approach on interbank interest rates (a simulated Euribor). In this sense the synthetic time series generated can mimic some anomalies due eventually to fraud over time. We simulate in specific dates, anomalies in the interbank interest rate data. The simulation: Anomaly Detection from the fitted models we are able to identifying the different anomalies which can occur over time. In this paper, anomaly detection allowed us to find, in a certain series of data, errors that could be attributed to a specific event that caused interbank interest rate to rise.

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

Anomaly Detection

Simulation Study

We consider various different simulations of interest rates. In particular each interest rate simulation is characterised by 200 daily observations over time. For each different time series we consider some structured shocks we try to detects using the techniques considered We follow all the different steps of the strategy considered. Finally we detects the different anomalies on the interest rates over time.

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

Anomaly Detection

Simulation Study

Auto-Arima Model Test set

ME 0.09

RMSE 0.09

MAE 0.09

MPE -4.70

MAPE 4.70

Test set

ME 0.13

RMSE 0.13

MAE 0.13

MPE -7.38

MAPE 7.38

ANN Model

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

Simulation Study

Anomaly Detection

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

Anomaly Detection

Simulation Study

Auto-Arima Model Test set

ME 0.29

RMSE 0.29

MAE 0.29

MPE 8.61

MAPE 8.61

Test set

ME 0.22

RMSE 0.22

MAE 0.22

MPE 6.74

MAPE 6.74

ANN Model

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

Simulation Study

Anomaly Detection

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

Anomaly Detection

Simulation Study

The hybrid forecasting approach: Test set

ME 0.00

RMSE 2.07

MAE 1.17

MPE 3.34

MAPE 10.76

ACF1 -0.13

Theil’s U 0.75

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

Anomaly Detection

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

Anomaly Detection

Conclusions

In this work we have considered the approach of hybrid forecasting (Zhang 2003) on the context of the anomaly detection (De Nadai Van Someren 2015). The advantage on using this approach on the interest rates are twofold: We decompose the interest rates into a nonlinear and a linear part We consider a distinct model for the two components The advantage is that the approach when the forecasting model can be considered good can discriminate the ”signal” from the ”noise” of the time series. At this point it is possible to identify the anomalies.

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Hybrid Forecasting Anomaly Detection and Interbank Interest Rates

Anomaly Detection

References

Bianchetti, M. (2011). The Zeeman Effect in Finance: Libor Spectroscopy and Basis Risk Management. Working paper Brutti, N. (2013). La Manipolazione Degli Indici Finanziari: Un Illecito in Cerca Di Identità (Manipulation of Financial Indices: A Wrongful Act in Search for Identity). Brousseau, V., Chailloux, A., & Durré, A. (2009). Interbank offered rate: Effects of the financial crisis on the information content of the fixing. Lille Economie and Management Working Paper, 17. Chandola, V., Banerjee, A., & Kumar, V. (2009). Anomaly detection: A survey. ACM computing surveys (CSUR), 41(3), 15. Chandola, V., Cheboli, D., & Kumar, V. (2009). Detecting anomalies in a time series database. Department of Computer Science and Engineering, University of Minnesota, Tech. Rep. TR, 09-004.

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Anomaly Detection

References

Chu C S J, Stinchcombe M, White H (1996). Monitoring Structural Change. Econometrica,64(5), 1045(1065). Cheboli, D. (2010). Anomaly detection of time series (Doctoral dissertation, University of Minnesota). Gordon, A.D. & Birks, H.J.B. (1972) Numerical methods in Quaternary palaeoecology I. Zonation of pollen diagrams. New Phytologist, 71, 961-979. De Nadai, M., & van Someren, M. (2015, July). Short-term anomaly detection in gas consumption through arima and artificial neural network forecast. In Environmental, Energy and Structural Monitoring Systems (EESMS), 2015 IEEE Workshop on (pp. 250-255). IEEE. Di Fonzo, T., & Lisi, F. (2005). Serie storiche economiche: analisi statistiche e applicazioni. Carocci. Diday, E., & Noirhomme-Fraiture, M. (Eds.). (2008). Symbolic data analysis and the SODAS software (pp. 187-188). J. Wiley & Sons. Eisl, A., Jankowitsch, R., & Subrahmanyam, M. G. (2014). The manipulation potential of Libor and Euribor.

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Anomaly Detection

References

Erturk, I., & Gabor, D. (Eds.). (2016). The Routledge Companion to Banking Regulation and Reform. Routledge. Fox, A. J. (1972). Outliers in time series.J. Royal Statis. Soc. Series B 34, 3, 350?363 Frunza, M. (2013). Market Manipulation and Moral Hazard: Can the LIBOR be Fixed?. Garziano G. (2016) A Brief Survey of R Packages: Anomaly Detection in Time Series Data. Around R http://aroundr.blogspot.it/2016/10/anomaly-detection-in-time-series-data.html Gaspar, V., Pérez-Quirós, G., & Rodríguez Mendizábal, H. (2004). Interest rate determination in the interbank market. 7-11

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Anomaly Detection

References

Goldberg, D., & Shan, Y. (2015). The importance of features for statistical anomaly detection. In 7th USENIX Workshop on Hot Topics in Cloud Computing (HotCloud 15). Hodge, V. J.; Austin, J. (2004). A Survey of Outlier Detection Methodologies. Artificial Intelligence Review. 22 (2): 85-126 Hou, D., & Skeie, D. R. (2014). LIBOR: origins, economics, crisis, scandal, and reform. LIBOR Usage and Substitutes, 2. Hyndman R.J. (2017). forecast: Forecasting functions for time series and linear models. R package version 8.0, URL:http://github.com/robjhyndman/forecast.

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Anomaly Detection

References

Hyndman R.J. and Khandakar Y (2008). ”Automatic time series forecasting: the forecast package for R.” Journal of Statistical Software, 26 (3), pp. 1-22. URL: http://www.jstatsoft.org/article/view/v0270i03. Hyndman, R. J., Wang, E., & Laptev, N. (2015, November). Large-scale unusual time series detection. In 2015 IEEE International Conference on Data Mining Workshop (ICDMW) (pp. 1616-1619). IEEE. Lopes D. S., Erturk, I., & Gabor, D. (Eds.). (2016). The Routledge Companion to Banking Regulation and Reform. Libor and Euribor, from normal banking practice to manipulation to the potential reform, 225-226 Lotze, T. H. (2009). Anomaly detection in time series: Theoretical and practical improvements for disease outbreak detection. Pimenta, C. (2014). Chapter one Notes on the Epistemology of Fraud in Pimenta C. Afonso O. Interdisciplinary Insights on Fraud, 17, 8.

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Anomaly Detection

References

Rahm, E., & Do, H. H. (2000). Data cleaning: Problems and current approaches. IEEE Data Eng. Bull., 23(4), 3-13. Rauch, B., Goettsche, M., & El Mouaaouy, F. (2013). LIBOR Manipulation?Empirical Analysis of Financial Market Benchmarks Using Benford’s Law. Siviero L. e Pellegrini F.M. (2014). La truffa del LIBOR e dell’EURIBOR Vallis, O., Hochenbaum, J., & Kejariwal, A. (2014). A novel technique for long-term anomaly detection in the cloud. In 6th USENIX Workshop on Hot Topics in Cloud Computing (HotCloud 14).

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Anomaly Detection

References

Verbesselt J, Hyndman R, Newnham G, Culvenor D (2010a). Detecting Trend and Seasonal Changes in Satellite Image Time Series. Remote Sensing of Environment, 114(1), 106-115. doi:10.1016/j.rse.2009.08.014 Verbesselt J, Hyndman R, Zeileis A, Culvenor D (2010b). Phenological Change Detection while Accounting for Abrupt and Gradual Trends in Satellite Image Time Series. Remote Sensing of Environment,114(12), 2970(2980) Verbesselt, J., Zeileis, A., & Herold, M. (2012). Near real-time disturbance detection using satellite image time series. Remote Sensing of Environment, 123, 98-108. Vishwasrao, S. et al. (2016) Anomaly Detection for Univariate Time-Series Data. Working Paper. Zhang, G. P. (2003). Time series forecasting using a hybrid ARIMA and neural network model.Neurocomputing, 50, 159-175.

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