research article the outlier interval detection algorithms...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 979035 6 pageshttpdxdoiorg1011552013979035

Research ArticleThe Outlier Interval Detection Algorithms on AstronauticalTime Series Data

Wei Hu and Junpeng Bao

School of Electronic and Information Engineering Xirsquoan Jiaotong University Xirsquoan 710049 China

Correspondence should be addressed to Junpeng Bao baojpmailxjtueducn

Received 6 November 2012 Accepted 11 February 2013

Academic Editor Tsung-Chih Lin

Copyright copy 2013 W Hu and J Bao This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The Outlier Interval Detection is a crucial technique to analyze spacecraft fault locate exception and implement intelligent faultdiagnosis system The paper proposes two OID algorithms on astronautical Time Series Data that is variance based OID (VOID)and FFT and 119896 nearest Neighbour based OID (FKOID) The VOID algorithm divides TSD into many intervals and measures eachintervalrsquos outlier score according to its variance This algorithm can detect the outlier intervals with great fluctuation in the timedomain It is a simple and fast algorithm with less time complexity but it ignores the frequency informationThe FKOID algorithmextracts the frequency information of each interval by means of Fast Fourier Transform so as to calculate the distances betweenfrequency features and adopts theKNNmethod tomeasure the outlier score according to the sumof distances between the intervalrsquosfrequency vector and the 119870 nearest frequency vectors It detects the outlier intervals in a refined way at an appropriate expense ofthe time and is valid to detect the outlier intervals in both frequency and time domains

1 Introduction

TheTime SeriesData is a sequence of values observed in someperiods Usually it takes a long time to continuously observean object and record its data so the accumulated TSD is oftenin a very large amount A significant issue is that how tominelatent or interesting knowledge from the huge amount of TSDand apply what is mined to the futureThe Astronautical data(AD) is a typical big TSD which is gathered by frequentlyand continuously observing spacecrafts The AD miningtechniques have many significant applications including butnot limited to analyzing satellitesrsquo working status judgingfaultserrors and forecasting its next state in the near futureSince it is hard to touch or measure spacecrafts in a shoutingdistance The technique to detect outlier intervals from ADis important The outlier intervals are the exceptional orunusual parts in a long period of the TSD which cover abun-dant information about spacecraft faults or special eventsTheOutlier Interval Detection technique can provide foundationproofs for intelligent astronautical fault analysis diagnosisand prediction

Althoughmost spacecraftsworkwell inmost of their livesthe faults and abnormal situations take place occasionally

Namely the outlier intervals are always drowned in longpieces of regular data Obviously it is a very toughmission fora person to find out outliers and make out regulations fromthe huge AD Human experts will spend not only plenty ofefforts and time but also some luck This paper proposestwo algorithms to automatically detect the outlier intervalsin AD by means of data mining These algorithms not onlyare able to promote analysis efficiency raise diagnosis systemrsquosresponse performance but also can be used to support space-craft fault prediction and diagnosis system as well as astro-nautical intelligent monitoring system Additionally the OIDtechnique has a great perspective in the area of industry pro-duction process monitoring financial administration medi-cal data analysis disaster alarm network intrusion detectioncredit card fraud detection and so forth

2 Related Work

Generally there are four ways to implement Outlier IntervalDetection that is distance based approach statistic basedapproach deviation based approach and clustering basedapproach

2 Mathematical Problems in Engineering

0 2 4 6 8 10 120

1

2

3

4

Time (months)

Valu

e

larr Outlier intervalsOutlier intervalsrarr

Figure 1 The examples of the outlier interval

(1) The distance based OID calculates the distancebetween objects firstly and then the outliers are defined as theobjects whose distances to others exceed the given threshold[1] This is a popular approach because it is simple and easyto implement and does require the knowledge of the datadistribution in advance

Angiulli et al have done a lot of research on this way [2ndash5] Angiulli and Pizzuti [2] defined an objectrsquos outlier scoreas the sum of distances between the object and its 119870 nearestneighborsThey also introduced a notion of outlier detectionsolving set [3] a subset of the data set to improve the distancebased outlier detection performance Angiulli and Fassetti [4]proposed a DOLPHIN algorithm to carry out the distancebased outlier detection in the large database Later they pro-posed a new model [5] that considers only the sum of thedistances between the object and the others in the slidingwindow

Chen et al [6] proposed a rough set and119870Nearest Neigh-bor (KNN) based method to detect the outliers on the mixedcontinuous and discrete dataset Bhaduri and Matthews[7] introduced two distributed algorithms and a novel index-ing scheme to speed up the distance based outlier detectionLi et al [8] transformed interval values into real values andthen employed the KNN approach to find outliers

(2) The statistic based OID approach requires the knowl-edge the probability distribution of the data in advance whichis the foundation of the approach But for a specific samplespace it is usually hard to know the exact distribution of thedata The key work of the approach is to perform plenty oftests in order to get the most proper distribution model Forexample Takeuchi and Yamanishi [9] proposed a unifyingframework that employed a Gaussian mixture model forstatistical outlier detection

(3)The deviation based OID extracts the main features ofthe center objects and then the outliers are considered as theobjects whose features deviate from the centers remarkablyFor example Oliveira and Meira [10] proposed a neural net-workmethod to forecast the thresholds for detecting outliers

(4)The clustering based OID considers the outliers as theby-products of clustering [11ndash13] that is the objects that donot belong to any normal cluster

Some OID methods exploit the Fourier Transform andthe Wavelet Transform to extract data features and mine theoutliers in a special domain For example Rasheed et al [14]

proposed a Fast Fourier Transform and Inverse Fast FourierTransform based OIDmethod Grane and Veiga [15] proposea Wavelet Transform based method to detect outliers infinancial data

3 The Outlier Interval Detection

31 The Solo Variant TSD and the Outlier Interval The solovariant TSD is the observed value of the single object orattribute in a period Generally a long period of TSD isdivided into many intervals according to the time scale Thusan interval is a piece of time and the values in the range Thetime span of each interval can be equal to each other suchas a day a week or 10 days Some applications may employunequal interval But in this paper a TSD is divided intoidentical length of intervals

The OID of the solo variant aims to find out the oddest119870 outlier intervals from a long period of solo variant TSDsuch as the changes of a satellite batteryrsquos voltage in a year Anoutlier interval indicates that the variant varies abnormallyin the time span so that it deserves a special care It oftenreflects some events happened or the symptoms of a faultThereason of the abnormal data may be diverse such as devicefault external interference changes of temperature and soforth Figure 1 illustrates some examples of the outlier intervalwhere the 3rd 4th 9th and 10th intervals are apparently dif-ferent from others

32 The Variance Based Outlier Interval Detection In a realastronautical dataset the abnormally varying data often resultin a great fluctuation amplitude The degree of the amplitudecan be directly reflected by the variance in the intervalNamely the outlier score can be measured by the varianceThe higher the value of the variance is the odder the intervalIt is easy to mine the top 119870 oddest intervals by means ofsorting their variances in descending order

The time complexity of the standard variance definition is119874(1198732) where 119873 is the number of data because it traverses

twice the whole sequence For a huge amount of AD it isbetter to execute a faster algorithm with a smaller time complexity We transform the standard variance defination to theform given in (1) Both results are identical but (1) needs only

Mathematical Problems in Engineering 3

0 50 1000

500

1000 Interval 1

0 50 1000

500

1000 Interval 2

0 50 1000

500

1000 Interval 3

0 50 1000

500

1000 Interval 4

0 50 1000

500

1000 Interval 5

0 50 1000

500

1000 Interval 6

0 50 1000

500

1000 Interval 7

0 50 1000

500

1000 Interval 8

0 50 1000

500

1000 Interval 9

0 50 1000

500

1000Interval 10

0 50 1000

500

1000 Interval 11

0 50 1000

500

1000 Interval 12

Figure 2 The frequency spectra of 12 intervals of the data shown in the Figure 1

Input119883 is the TSD and 119888 is the count of the intervalsOutput The top 119870 outlier intervals in time domain and their scores

(1) 119879 is the set of 119888 intervals divided from119883 that is 119879 = 119905119894| ⋃119888

119894=1119905119894= 119883 119905

119894cap 119905119895= 120601(119894 = 119895)

(2) for each 119905119894in 119879

(3) 119886 = sum119909119895isin119905119894

1199092

119895 119887 = sum

119909119895isin119905119894119909119895

(4) var(119905119894) =

119886

|119905119894|minus (

11988710038161003816100381610038161199051198941003816100381610038161003816

)

2

(5) endfor(6) Outlier(119879119870) = argmax119870

119905119894isin119879var(119905

119894)

The time complexity of the VOID is 119874(119873) where119873 is the number of data in the TSD

Pseudocode 1

once traverse so its time complexity is 119874(119873) one has

var = 1119899

119899

sum

119894=1

1199092

119894minus (119909)2 (1)

Pseudocode 1 is the pseudo code of the VOID

33 FFT and K Nearest Neighbour Based Outlier IntervalDetection The VOID algorithm can quickly detect the out-lier intervals in time domainHowevermany real AD are per-iodical in some extentThe violent frequency fluctuations alsoimply something happened On the other hand the violentfluctuations in time domain lead to changes in frequencydomain Figure 2 illustrates the frequency spectra of thetwelve intervals of the data shown in Figure 1 It is clear that

the frequency fluctuations in the 3rd 4th 9th and 10th inter-vals are distinctly greater than others

In order to detect the outliers in a fine granularity morefrequencies have to be taken into account So a feature vectorof an interval is made of the whole frequency band from thelowest frequency to the highest The outlier score of an inter-val is measured according to the distance between featurevectors instead of varianceMoreover an amplitude thresholdis set to decline noises namely the value is assigned to 0 if itis not higher than the threshold otherwise it keeps its value

The FKOID algorithm firstly divides the whole TSD into119888 intervals equally Secondly it executes FFT on each intervalto get the frequency spectrum of the interval which buildsa feature vector after the low energy frequencies are set to 0by the amplitude threshold Thirdly the Euclidean distancesbetween every pair of feature vectors are calculated It should


0 2 4 6 8 10 120

2

4

Valu

e

Δ Δ Δ Δ

Time (months)

(a)

0 2 4 6 8 10 1230

35

40

Valu

e

Δ Δ Δ Δ

Time (months)

(b)

0 2 4 6 8 10 12

0204060

Valu

e

minus40minus20

Δ Δ Δ Δ

Time (months)

(c)

0 2 4 6 8 100

204060

Valu

e

Δ Δ Δ Δ

Time (months)

(d)

0 2 4 6 8 10Time (days)

Valu

e

02

0

minus02

ΔΔ Δ Δ

(e)

0 2 4 6 8 10

0

Valu

eminus5

minus10

Δ Δ Δ Δ

Time (days)

(f)

Figure 3 The VOID results on the 6 real astronautical data where the symbol Δmarks the outlier intervals

Table 1 The top 4 outlier scores of both algorithms on the 6 real astronautical datasets where the number in the brackets is the ID of theinterval

Algorithm (a) (b) (c) (d) (e) (f)

VOIDFigure 3

12001(3) 17812(4) 13166(3) 13583(5) 00076(5) 54093(10)10300(9) 16907(9) 7706(10) 12809(10) 00062(1) 53548(7)06976(10) 14850(3) 7281(9) 12737(3) 00055(4) 53369(8)03677(4) 13747(10) 2137(2) 11052(9) 00055(9) 51904(9)

FKOIDFigure 4

7211(3) 9013(3) 28163(3) 39757(5) 6094(4) 1479(1)6515(9) 8856(9) 28083(9) 39619(2) 5502(1) 1306(10)6027(10) 7145(10) 26842(10) 39182(1) 5310(5) 1149(7)4862(4) 5746(4) 18941(2) 38768(4) 5303(9) 1096(2)

be noted that some frequencies may have an extremely largeamplitude so that they are overwhelming in the vector Thatwill cover the other frequenciesrsquo effect in terms of Euclideandistance So we set a top threshold to limit the maximumamplitude value of those overwhelming frequencies At lastthe outlier score of an interval is the sumof distances betweenthe intervalrsquos feature vector and its 119870 nearest neighbors

The idea of the FKOID is inspired by the method ofGrane and Veiga [15] The longer the vectorrsquos distance is theodder the interval is Pseudocode 2 is the pseudo code of theFKOID

The FKOID algorithm has to maintain a distance matrixand fetches the 119896 nearest neighbors of each interval Thetime complexity of that is 119874(1198961199081198882) where 119896 is the numberof nearest neighbors 119908 is the number of data in an interval

and 119888 is the count of intervals Since the first stage is the FastFourier Transform of each interval the entire time complex-ity of the FKOID is 119874(119873 times lg(119908) + 1198961199081198882)

4 Experimental Results

We use six real astronautical datasets to test the algorithmsAs shown in Figure 3 these AD represent some popular ten-dencies in the real world The data of (a) (b) and (c)have distinct violent fluctuations in time domain The dataof (e) and (f) have no great fluctuation in time domainThe data of (d) is the most complicated which containsdiverse variation tendency Figure 3 illustrate the top 4 outlierintervals detected by the VOID algorithm and Figure 4 is


0 2 4 6 8 10 120

2

4

Valu

e

nablanablanablanabla

Time (months)

(a)

0 2 4 6 8 10 1230

35

40

Valu

e

Time (months)


(b)

0 2 4 6 8 10 12

0204060

Valu

e

minus40minus20


Time (months)

(c)

0 2 4 6 8 100

204060

Valu

e


Time (months)

(d)

0 2 4 6 8 10

Valu

e

02

0

minus02


Time (days)

(e)

0 2 4 6 8 10

0

Valu

eminus5

minus10


Time (days)

(f)

Figure 4 The FKOID results on the 6 real astronautical data where the symbol nablamarks the outlier intervals

Input119883 is the TSD and 119888 is the count of the intervals [1205911 1205912] are the amplitude thresholds

Output The top 119870 outlier intervals and their scores(1) 119879 is the set of 119888 intervals divided from119883 that is 119879 = 119905

119894| ⋃119888

119894=1119905119894= 119883 119905

119894cap 119905119895= 120601(119894 = 119895)

(2) for each 119905119894in 119879

(3) 119891119905119894= FFT(119905

119894) otimes [120591

1 1205912] where 119886 otimes [120591

1 1205912] =

1205912 119886 ge 120591

2

119886 1205911lt 119886 lt 120591

2

0 119886 le 1205911

(4) endfor(5) for each 119905

119894in 119879

(6) 119889(119905119894) = sum

119891119905119895isin119896119899119899(119891119905119894

)|119891119905119894minus 119891119905119895|2

(7) endfor(8) Outlier(119879 119870) = argmax119870

119905119894isin119879119889(119905119894)

Pseudocode 2

the results of FKOID respectively The Table 1 lists the de-tailed outlier scores on the 6 AD

Based on the experimental results the VOID algorithm isfit for the case that data varies slightly in the regular situationwhereas it becomes violent in the irregular situation If thenormal data waves frequently and varies widely then theVOID algorithm will have a great error Additionally theVOID algorithm is suitable to detectOID in time domain butfailed in the case that data varies peacefully in time domainbut violently in frequency domain

The FKOID algorithm can solve the problem of OID infrequency domain It is shown in Figure 4 that the violentfluctuations in time domain have corresponding abnormal

changes in the frequency domain As a result the FKOIDalgorithm can detect outlier intervals in time domain as wellHowever the FKOID algorithm can detect the very subtleoutliers at the expense of long running time because it hasa high time complexity

5 Conclusions

The OID technique can quickly deal with TSD to find theoddest objects which often imply crucial exceptional eventsIt has a great perspective in the astronautical applicationssuch as the spacecraft fault prediction and diagnosis system


astronautical intelligentmonitoring systemandother systemsbased on TSD

This paper proposes two algorithms to detect the outlierintervals on astronautical data The VOID algorithm directlyexploits the variance of data to quickly detect the outlier inter-vals in time domain The FKOID employ the full frequencyband to build a feature vector of an interval and measure theoutlier score by the distances sum of the119870 nearest neighborsThe FKOID is subtle enough to detect the outliers in a refinedgranularity in both frequency and time domains but its timecomplexity is a little big

However the above algorithms are based on the identicallength of interval It is rather arbitrary in practice because thereal outlier intervalsmay be varying in length So it is our nextwork to study on the methods of the unequal length OutlierInterval Detection

Acknowledgments

This research is supported by National Natural ScienceFoundation of China (Grant 60903123) and the BaiduThemeResearch Plan on Large Scale Machine Learning and DataMining

References

[1] E Knorr and R Ng ldquoTucakov Distance-based outliers algo-rithms and applicationsrdquo VLDB Journal vol 8 no 3 pp 237ndash253 2000

[2] F Angiulli and C Pizzuti ldquoOutlier mining in large high-dimen-sional data setsrdquo IEEE Transaction on Knowledge and DataEngineering vol 17 no 2 pp 203ndash215 2005

[3] F Angiulli S Basta and C Pizzuti ldquoDistance-based detectionand prediction of outliersrdquo IEEETransactions onKnowledge andData Engineering vol 18 no 2 pp 145ndash160 2006

[4] FAngiulli and F Fassetti ldquoAn efficient algorithm formining dis-tance-based outliers in very large datasetsrdquo ACM Transactionson Knowledge Discovery from Data vol 3 no 1 pp 1ndash57 2009

[5] F Angiulli and F Fassetti ldquoDistance-based outlier queries indata streams the novel task and algorithmsrdquo Data Mining andKnowledge Discovery vol 20 no 2 pp 290ndash324 2010

[6] Y Chen D Miao and H Zhang ldquoNeighborhood outlier detec-tionrdquo Expert Systems with Applications vol 37 no 12 pp 8745ndash8749 2010

[7] K Bhaduri andB LMatthews ldquoAlgorithms for speeding up dis-tance-based outlier detectionrdquo in Proceedings of the 17th ACMSIGKDD International Conference on Knowledge Discovery andData Mining (SIGKDD rsquo11) pp 859ndash867 2011

[8] S Li R Lee and S D Lang ldquoDetecting outliers in intervaldatardquo in Proceedings of the Southeast regional conference (ACM-SE rsquo06) pp 290ndash295 2006

[9] J Takeuchi andKYamanishi ldquoAunifying framework for detect-ing outliers and change points from non-stationary time seriesdatardquo IEEE Transactions on Knowledge and Data Engineeringvol 18 no 4 pp 482ndash492 2006

[10] A Oliveira and S Meira ldquoDetecting novelties in time seriesthrough neural networks forecasting with robust confidenceintervalsrdquo Neurocomputing vol 70 no 1ndash3 pp 79ndash92 2006

[11] C Bohm C Faloutsos and C Plant ldquoOutlier-robust clusteringusing independent componentsrdquo in Proceedings of the 28th

ACM SIGMOD International Conference on Management ofData (SIGMOD rsquo08) pp 185ndash198 2008

[12] N Ade and B Zadronzy ldquoOutlier detection by active learningrdquoin Proceedings of the 12th ACM SIGMOD International Confer-ence onManagement of Data (SIGMOD rsquo06) pp 504ndash509 2006

[13] C Franke andM Gertz ldquoORDEN outlier region detection andexploration in sensor networksrdquo in Proceedings of the 29th ACMSIGMOD International Conference on Management of Data(SIGMOD rsquo09) pp 1075ndash1078 2009

[14] F Rasheed et al ldquoFourier transform based spatial outlierminingrdquo Lecture Notes in Computer Science pp 317ndash324 2009

[15] A Grane and H Veiga ldquoWavelet-based detection of outliers infinancial time seriesrdquo Computational Statistics amp Data Analysisvol 54 no 11 pp 2580ndash2593 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


0 2 4 6 8 10 120

1

2

3

4

Time (months)

Valu

e

larr Outlier intervalsOutlier intervalsrarr

Figure 1 The examples of the outlier interval

(1) The distance based OID calculates the distancebetween objects firstly and then the outliers are defined as theobjects whose distances to others exceed the given threshold[1] This is a popular approach because it is simple and easyto implement and does require the knowledge of the datadistribution in advance

Angiulli et al have done a lot of research on this way [2ndash5] Angiulli and Pizzuti [2] defined an objectrsquos outlier scoreas the sum of distances between the object and its 119870 nearestneighborsThey also introduced a notion of outlier detectionsolving set [3] a subset of the data set to improve the distancebased outlier detection performance Angiulli and Fassetti [4]proposed a DOLPHIN algorithm to carry out the distancebased outlier detection in the large database Later they pro-posed a new model [5] that considers only the sum of thedistances between the object and the others in the slidingwindow

Chen et al [6] proposed a rough set and119870Nearest Neigh-bor (KNN) based method to detect the outliers on the mixedcontinuous and discrete dataset Bhaduri and Matthews[7] introduced two distributed algorithms and a novel index-ing scheme to speed up the distance based outlier detectionLi et al [8] transformed interval values into real values andthen employed the KNN approach to find outliers

(2) The statistic based OID approach requires the knowl-edge the probability distribution of the data in advance whichis the foundation of the approach But for a specific samplespace it is usually hard to know the exact distribution of thedata The key work of the approach is to perform plenty oftests in order to get the most proper distribution model Forexample Takeuchi and Yamanishi [9] proposed a unifyingframework that employed a Gaussian mixture model forstatistical outlier detection

(3)The deviation based OID extracts the main features ofthe center objects and then the outliers are considered as theobjects whose features deviate from the centers remarkablyFor example Oliveira and Meira [10] proposed a neural net-workmethod to forecast the thresholds for detecting outliers

(4)The clustering based OID considers the outliers as theby-products of clustering [11ndash13] that is the objects that donot belong to any normal cluster

Some OID methods exploit the Fourier Transform andthe Wavelet Transform to extract data features and mine theoutliers in a special domain For example Rasheed et al [14]

proposed a Fast Fourier Transform and Inverse Fast FourierTransform based OIDmethod Grane and Veiga [15] proposea Wavelet Transform based method to detect outliers infinancial data

3 The Outlier Interval Detection

31 The Solo Variant TSD and the Outlier Interval The solovariant TSD is the observed value of the single object orattribute in a period Generally a long period of TSD isdivided into many intervals according to the time scale Thusan interval is a piece of time and the values in the range Thetime span of each interval can be equal to each other suchas a day a week or 10 days Some applications may employunequal interval But in this paper a TSD is divided intoidentical length of intervals

The OID of the solo variant aims to find out the oddest119870 outlier intervals from a long period of solo variant TSDsuch as the changes of a satellite batteryrsquos voltage in a year Anoutlier interval indicates that the variant varies abnormallyin the time span so that it deserves a special care It oftenreflects some events happened or the symptoms of a faultThereason of the abnormal data may be diverse such as devicefault external interference changes of temperature and soforth Figure 1 illustrates some examples of the outlier intervalwhere the 3rd 4th 9th and 10th intervals are apparently dif-ferent from others

32 The Variance Based Outlier Interval Detection In a realastronautical dataset the abnormally varying data often resultin a great fluctuation amplitude The degree of the amplitudecan be directly reflected by the variance in the intervalNamely the outlier score can be measured by the varianceThe higher the value of the variance is the odder the intervalIt is easy to mine the top 119870 oddest intervals by means ofsorting their variances in descending order

The time complexity of the standard variance definition is119874(1198732) where 119873 is the number of data because it traverses

twice the whole sequence For a huge amount of AD it isbetter to execute a faster algorithm with a smaller time complexity We transform the standard variance defination to theform given in (1) Both results are identical but (1) needs only


0 50 1000

500

1000 Interval 1

0 50 1000

500

1000 Interval 2

0 50 1000

500

1000 Interval 3

0 50 1000

500

1000 Interval 4

0 50 1000

500

1000 Interval 5

0 50 1000

500

1000 Interval 6

0 50 1000

500

1000 Interval 7

0 50 1000

500

1000 Interval 8

0 50 1000

500

1000 Interval 9

0 50 1000

500

1000Interval 10

0 50 1000

500

1000 Interval 11

0 50 1000

500

1000 Interval 12




119894=1119905119894= 119883 119905

119894cap 119905119895= 120601(119894 = 119895)

(2) for each 119905119894in 119879

(3) 119886 = sum119909119895isin119905119894

1199092

119895 119887 = sum

119909119895isin119905119894119909119895

(4) var(119905119894) =

119886

|119905119894|minus (

11988710038161003816100381610038161199051198941003816100381610038161003816

)

2


119905119894isin119879var(119905

119894)


Pseudocode 1


var = 1119899

119899

sum

119894=1

1199092

119894minus (119909)2 (1)







0 2 4 6 8 10 120

2

4

Valu

e

Δ Δ Δ Δ

Time (months)

(a)

0 2 4 6 8 10 1230

35

40

Valu

e

Δ Δ Δ Δ

Time (months)

(b)

0 2 4 6 8 10 12

0204060

Valu

e

minus40minus20

Δ Δ Δ Δ

Time (months)

(c)

0 2 4 6 8 100

204060

Valu

e

Δ Δ Δ Δ

Time (months)

(d)

0 2 4 6 8 10Time (days)

Valu

e

02

0

minus02

ΔΔ Δ Δ

(e)

0 2 4 6 8 10

0

Valu

eminus5

minus10

Δ Δ Δ Δ

Time (days)

(f)




VOIDFigure 3

12001(3) 17812(4) 13166(3) 13583(5) 00076(5) 54093(10)10300(9) 16907(9) 7706(10) 12809(10) 00062(1) 53548(7)06976(10) 14850(3) 7281(9) 12737(3) 00055(4) 53369(8)03677(4) 13747(10) 2137(2) 11052(9) 00055(9) 51904(9)

FKOIDFigure 4

7211(3) 9013(3) 28163(3) 39757(5) 6094(4) 1479(1)6515(9) 8856(9) 28083(9) 39619(2) 5502(1) 1306(10)6027(10) 7145(10) 26842(10) 39182(1) 5310(5) 1149(7)4862(4) 5746(4) 18941(2) 38768(4) 5303(9) 1096(2)








0 2 4 6 8 10 120

2

4

Valu

e


Time (months)

(a)

0 2 4 6 8 10 1230

35

40

Valu

e

Time (months)


(b)

0 2 4 6 8 10 12

0204060

Valu

e

minus40minus20


Time (months)

(c)

0 2 4 6 8 100

204060

Valu

e


Time (months)

(d)

0 2 4 6 8 10

Valu

e

02

0

minus02


Time (days)

(e)

0 2 4 6 8 10

0

Valu

eminus5

minus10


Time (days)

(f)




119894| ⋃119888

119894=1119905119894= 119883 119905

119894cap 119905119895= 120601(119894 = 119895)

(2) for each 119905119894in 119879

(3) 119891119905119894= FFT(119905

119894) otimes [120591

1 1205912] where 119886 otimes [120591

1 1205912] =

1205912 119886 ge 120591

2

119886 1205911lt 119886 lt 120591

2

0 119886 le 1205911


119894in 119879

(6) 119889(119905119894) = sum

119891119905119895isin119896119899119899(119891119905119894

)|119891119905119894minus 119891119905119895|2


119905119894isin119879119889(119905119894)

Pseudocode 2





5 Conclusions






Acknowledgments


References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 50 1000

500

1000 Interval 1

0 50 1000

500

1000 Interval 2

0 50 1000

500

1000 Interval 3

0 50 1000

500

1000 Interval 4

0 50 1000

500

1000 Interval 5

0 50 1000

500

1000 Interval 6

0 50 1000

500

1000 Interval 7

0 50 1000

500

1000 Interval 8

0 50 1000

500

1000 Interval 9

0 50 1000

500

1000Interval 10

0 50 1000

500

1000 Interval 11

0 50 1000

500

1000 Interval 12




119894=1119905119894= 119883 119905

119894cap 119905119895= 120601(119894 = 119895)

(2) for each 119905119894in 119879

(3) 119886 = sum119909119895isin119905119894

1199092

119895 119887 = sum

119909119895isin119905119894119909119895

(4) var(119905119894) =

119886

|119905119894|minus (

11988710038161003816100381610038161199051198941003816100381610038161003816

)

2


119905119894isin119879var(119905

119894)


Pseudocode 1


var = 1119899

119899

sum

119894=1

1199092

119894minus (119909)2 (1)







0 2 4 6 8 10 120

2

4

Valu

e

Δ Δ Δ Δ

Time (months)

(a)

0 2 4 6 8 10 1230

35

40

Valu

e

Δ Δ Δ Δ

Time (months)

(b)

0 2 4 6 8 10 12

0204060

Valu

e

minus40minus20

Δ Δ Δ Δ

Time (months)

(c)

0 2 4 6 8 100

204060

Valu

e

Δ Δ Δ Δ

Time (months)

(d)

0 2 4 6 8 10Time (days)

Valu

e

02

0

minus02

ΔΔ Δ Δ

(e)

0 2 4 6 8 10

0

Valu

eminus5

minus10

Δ Δ Δ Δ

Time (days)

(f)




VOIDFigure 3

12001(3) 17812(4) 13166(3) 13583(5) 00076(5) 54093(10)10300(9) 16907(9) 7706(10) 12809(10) 00062(1) 53548(7)06976(10) 14850(3) 7281(9) 12737(3) 00055(4) 53369(8)03677(4) 13747(10) 2137(2) 11052(9) 00055(9) 51904(9)

FKOIDFigure 4

7211(3) 9013(3) 28163(3) 39757(5) 6094(4) 1479(1)6515(9) 8856(9) 28083(9) 39619(2) 5502(1) 1306(10)6027(10) 7145(10) 26842(10) 39182(1) 5310(5) 1149(7)4862(4) 5746(4) 18941(2) 38768(4) 5303(9) 1096(2)








0 2 4 6 8 10 120

2

4

Valu

e


Time (months)

(a)

0 2 4 6 8 10 1230

35

40

Valu

e

Time (months)


(b)

0 2 4 6 8 10 12

0204060

Valu

e

minus40minus20


Time (months)

(c)

0 2 4 6 8 100

204060

Valu

e


Time (months)

(d)

0 2 4 6 8 10

Valu

e

02

0

minus02


Time (days)

(e)

0 2 4 6 8 10

0

Valu

eminus5

minus10


Time (days)

(f)




119894| ⋃119888

119894=1119905119894= 119883 119905

119894cap 119905119895= 120601(119894 = 119895)

(2) for each 119905119894in 119879

(3) 119891119905119894= FFT(119905

119894) otimes [120591

1 1205912] where 119886 otimes [120591

1 1205912] =

1205912 119886 ge 120591

2

119886 1205911lt 119886 lt 120591

2

0 119886 le 1205911


119894in 119879

(6) 119889(119905119894) = sum

119891119905119895isin119896119899119899(119891119905119894

)|119891119905119894minus 119891119905119895|2


119905119894isin119879119889(119905119894)

Pseudocode 2





5 Conclusions






Acknowledgments


References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 2 4 6 8 10 120

2

4

Valu

e

Δ Δ Δ Δ

Time (months)

(a)

0 2 4 6 8 10 1230

35

40

Valu

e

Δ Δ Δ Δ

Time (months)

(b)

0 2 4 6 8 10 12

0204060

Valu

e

minus40minus20

Δ Δ Δ Δ

Time (months)

(c)

0 2 4 6 8 100

204060

Valu

e

Δ Δ Δ Δ

Time (months)

(d)

0 2 4 6 8 10Time (days)

Valu

e

02

0

minus02

ΔΔ Δ Δ

(e)

0 2 4 6 8 10

0

Valu

eminus5

minus10

Δ Δ Δ Δ

Time (days)

(f)




VOIDFigure 3

12001(3) 17812(4) 13166(3) 13583(5) 00076(5) 54093(10)10300(9) 16907(9) 7706(10) 12809(10) 00062(1) 53548(7)06976(10) 14850(3) 7281(9) 12737(3) 00055(4) 53369(8)03677(4) 13747(10) 2137(2) 11052(9) 00055(9) 51904(9)

FKOIDFigure 4

7211(3) 9013(3) 28163(3) 39757(5) 6094(4) 1479(1)6515(9) 8856(9) 28083(9) 39619(2) 5502(1) 1306(10)6027(10) 7145(10) 26842(10) 39182(1) 5310(5) 1149(7)4862(4) 5746(4) 18941(2) 38768(4) 5303(9) 1096(2)








0 2 4 6 8 10 120

2

4

Valu

e


Time (months)

(a)

0 2 4 6 8 10 1230

35

40

Valu

e

Time (months)


(b)

0 2 4 6 8 10 12

0204060

Valu

e

minus40minus20


Time (months)

(c)

0 2 4 6 8 100

204060

Valu

e


Time (months)

(d)

0 2 4 6 8 10

Valu

e

02

0

minus02


Time (days)

(e)

0 2 4 6 8 10

0

Valu

eminus5

minus10


Time (days)

(f)




119894| ⋃119888

119894=1119905119894= 119883 119905

119894cap 119905119895= 120601(119894 = 119895)

(2) for each 119905119894in 119879

(3) 119891119905119894= FFT(119905

119894) otimes [120591

1 1205912] where 119886 otimes [120591

1 1205912] =

1205912 119886 ge 120591

2

119886 1205911lt 119886 lt 120591

2

0 119886 le 1205911


119894in 119879

(6) 119889(119905119894) = sum

119891119905119895isin119896119899119899(119891119905119894

)|119891119905119894minus 119891119905119895|2


119905119894isin119879119889(119905119894)

Pseudocode 2





5 Conclusions






Acknowledgments


References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 2 4 6 8 10 120

2

4

Valu

e


Time (months)

(a)

0 2 4 6 8 10 1230

35

40

Valu

e

Time (months)


(b)

0 2 4 6 8 10 12

0204060

Valu

e

minus40minus20


Time (months)

(c)

0 2 4 6 8 100

204060

Valu

e


Time (months)

(d)

0 2 4 6 8 10

Valu

e

02

0

minus02


Time (days)

(e)

0 2 4 6 8 10

0

Valu

eminus5

minus10


Time (days)

(f)




119894| ⋃119888

119894=1119905119894= 119883 119905

119894cap 119905119895= 120601(119894 = 119895)

(2) for each 119905119894in 119879

(3) 119891119905119894= FFT(119905

119894) otimes [120591

1 1205912] where 119886 otimes [120591

1 1205912] =

1205912 119886 ge 120591

2

119886 1205911lt 119886 lt 120591

2

0 119886 le 1205911


119894in 119879

(6) 119889(119905119894) = sum

119891119905119895isin119896119899119899(119891119905119894

)|119891119905119894minus 119891119905119895|2


119905119894isin119879119889(119905119894)

Pseudocode 2





5 Conclusions






Acknowledgments


References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Acknowledgments


References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article the outlier interval detection algorithms...

Documents