Performance analysis and implementation of an adaptive real-time weather forecasting system

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ContentslistsavailableatScienceDirect

Internet of Things

journalhomepage:www.elsevier.com/locate/iot

Performance analysis and implementation of an adaptive real-time weather forecasting system

T.P. Fowdur

^a

, Y. Beeharry

^a^,^∗

, V. Hurbungs

^b

, V. Bassoo

^a

, V. Ramnarain-Seetohul

^c

, E. Chan Moo Lun

^a

aDepartment of Electrical and Electronic Engineering , Faculty of Engineering, University of Mauritius, Mauritius

bDepartment of Software and Information Systems, Faculty on Information, Communication, and Digital Technologies, University of Mauritius, Mauritius

cDepartment of Information, Communication and Technology, Faculty of Information, Communication, and Digital Technologies, University of Mauritius, Mauritius

a rt i c l e i n f o

Article history:

Received 15 August 2018 Revised 31 August 2018 Accepted 1 September 2018 Available online 6 September 2018 Keywords:

Real-time weather forecasting Internet of things

K-nearest neighbors (K-NN) Multiple linear regression (MLR) Adaptive K-NN

Adaptive MLR

a b s t ra c t

Recently severalreal-timeweatherforecastingsystemsbasedonInternetofThings(IoT) havebeen developedtoprovideshort-termrealtimeforecastsalsoreferredtoas Now- casts.Themainchallengeinthesesystemsistouseappropriatepredictionalgorithmsthat canpredictdifferentweatherparameterswiththehighestpossibleaccuracy.Inthiswork, an IoT basedweather forecastingsystem has been implementedto provide short-term weatherforecastsonaUniversitycampusatintervalsrangingfrom20mintoonehour.

SeveraladaptiveforecastingalgorithmsbasedonvariantsoftheMultiple LinearRegres- siontechniqueaswellasK-NearestNeighbors(K-NN)havebeenexperimented.Moreover, threeadaptiveselectioncriteriaforselectingthemostappropriatepredictionalgorithmfor agivenNowcasthavebeendevelopedandtested.Theparametersanalysedare:temperature,humidity,atmosphericpressure,rainfall,luminosity,wind-speed,andwinddirection.

Thebestadaptiveschemes areabletopredicttheseparameterswithanoverallpercent- ageerrorof6.58%ascomparedtonon-adaptiveoneswhichpredictedwithaworstcase percentageerrorof13.59%.

1. Introduction

Weather forecastinghasalways beena vital processto ensure thesmooth running ofseveralimportant activities [1]. However,duetodrasticclimaticchanges,weatherpredictionusingthetraditionaltechniquesisbecomingineffective.Many countriesareatriskofflashfloodnowadaysandpredictingsuchweatherconditionswithconventionalforecastingsystems isnotpossiblebecausethesesystemsprovidepredictionsforlargeregionsoverhours. Mauritiushasrecentlyexperienced drasticweatherconditionssuchasflash-floodsthatcausedmajorcollateraldamageandlifeloss.However,nowadays,with theemergence ofInternet ofThings,realtimeornearrealtimeweather predictionispossible.Thereareseveralweather forecastingsystemsbasedonlocalizedsensorsconnectedtocloudcomputingfacilitiesthatcanprovideshort-termforecasts forsmallregions.Theseforecastingsystemsmake useoftechniquessuch asneuralnetworks, Fuzzylogic,time seriesand regressionanalysis.

∗ Corresponding author.

E-mail address: y.beeharry@uom.ac.mu (Y. Beeharry).

https://doi.org/10.1016/j.iot.2018.09.002

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ListofNotations

T TemperatureindegreesCelsius

H %Humidity

L Lightintensityin

R Rainfallinmm

P AtmosphericpressureinPa.

WD Winddirectionconvertedtoadegreesscale.

WS Windspeedinm/s

W Thetrainingwindowwhichwasﬁxedto120mininthiswork.

t Thetimeindex.

Pxt denotesanyotherrelatedparameterthatcouldinﬂuencethevalueofP1t. a₀(W),a₁(W)anda₂(W) are coeﬃcientsdetermined foratrainingwindowofsizeWwhichwaskept

at120min.

p representstheorderofautoregression,

q istheorderofmovingaverage(numberofpasterrorterms),

Ø istheslopecoeﬃcient,

Ɵ isthemovingaveragecoeﬃcient,

e istheerrorterm

μ isaconstantterm

K istheneighborhoodsize

y_k isthenearestreading

T_A Theactualorcurrenttime.

TS Thetimeatwhichtheﬁrstvalueinthetrainingwindowwasrecorded.

T_M Themid-timeintervalbetweenT_S andT_A.

T_A₊_I Thetimeatwhichthevaluesoftheparameteristobepredicted.Thesymbol IrepresentsanintervalinminutesaftertheactualtimeT_A.

W Thetrainingwindowsizeinminuteswhichessentiallyrepresentasetofpre- observed values ofthe parameter tobe predictedover thepast W minutes goingbackfromtheactualtimeT_AtoT₀.

V_A TheactualorcurrentvaluesoftheparameterbeingmeasuredattimeT_A. V_S ThevalueoftheparameterbeingmeasuredattimeTs.

V_M ThevalueoftheparameterbeingmeasuredattimeT_M.

Vˆ_A^x₊_I ThevalueoftheparameterpredictedwithP^x attimeT_A₊_I.

W/2 ItisequivalenttohalfthewindowsizeWandcontainsvaluesoftheparam- eterrecordedfromtimeT_M toT_A i.e.thevaluesV_MtoV_A.

Vˆ_M^x isavaluepredictedattime TMwithpredictorP^x usingavailablevaluesfrom V_StoV_M₋₁.

Vˆ_M^x₊₁ isavaluepredictedattimeT_M₊₁withpredictorP^xusingavailablevaluesfrom V_StoV_M.

Vˆ_M+^x _i isavaluepredictedattimeT_M₊_iwithpredictorP^xusingavailablevaluesfrom V_StoV_M₊_i₋₁.Theindexitakesvaluesfrom1toW/2.

SE_M^x,SE_M+1^x ,SE^x_M+2, ..., SE_M^x₊_i, ..., SE^x_A represent the squared Euclidean distances between the actual values V_M,V_M₊₁,V_M₊₂,….V_M₊_i…..V_A and Vˆ_M^x,Vˆ_M+1^x ,Vˆ_M+2^x , ..., Vˆ_M^x₊_i, ..., Vˆ_A^x respectively.

NumerousIoT basedsolutions havebeenimplemented toobserve weatherconditions atspecificlocations[2–5].Most systemsmonitoredenvironmentalconditionssuch astemperature, precipitation,relativehumidity,light intensityandCO₂ level.ThearchitecturesconsistedofaseriesofsensorsconnectedtoamicrocontrollerplatformsuchasArduinoorRaspberry Pi.Thedatacollectedweresent toeitheracomputeroracloud computingplatformforstorageandprocessing. Different methodswere usedto makethe dataavailable tousers ofthesystem. In[2],tweetswere usedto disseminatethe gath- eredinformationandfollowersofthetwitteraccountreceivedinformationregardingdaylight,temperatureandhumidity.In [6–8],flashfloods monitoringsystemwere implementedtoprovidetimelyinformationtothreatenedpopulationandcon- cernedauthoritieswhenthewaterlevelrisesbeyondthepredefinedthresholdvalue.SMSwasusedtosendalertmessages [6,8]. Waterquality classification using Pollution Indexmethod has been performed using linearSVM anddecision tree algorithms in[9] forthe preservation ofthe waterecosystem inmaritime andarchipelagic countries. A flood prediction systemwasalsoimplementedusingwaterlevelandvelocity ofthewater[6].In[10],theauthorsintroducedCloudCast,a mobileapplicationforlocalizedshort-termweatherprediction.CloudCastconsistsofanarchitecturelinkingweatherradars to cloud resources anda Nowcasting algorithm that accurately predicts short termweather conditions.According to the

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authors,CloudCastwasabletoperformaccuratepredictionwithlessthan2mindelaytodeliver15-minNowcastimageto amobileclient.

Severalalgorithms havebeenusedforweatherforecasting.Datacollectedfromsensorsarefed intoalgorithmstomake predictions.Thealgorithmsusedincluderegression–basedpredictionwherein[11]datawasgatheredlocallyatPantnagar Stationandprocessedtoobtainstatisticalmeasuresofthehiddeninformation.Furthermore,in[12]forecastingofseasonal andannualrainfallwasdonebyusinganonlinearmodelingwithGammaTest(GT)innorthofIran.Rainfallwasalsopre- dictedusingmodifiedlinearregressionindifferentregionsofSouthernIndiain[13].Also,Hadoop-basedARIMAalgorithm hasbeenusedtoforecastthedataofthecomingfortnightbytakingdatacollectedoverthepastdecadein[14].TheARIMA model wasalsoused toforecast wind speed foraperiod of7days inLatvia [15]andto forecastweather conditions for the AbadehregionofIran [15]. TheSeasonal ARIMAmodelwasused forforecastingofmonthlyrainfall andtemperature formonthlyintervalsforaregionofIndiain[16].In[17]theARIMAmodelwasappliedforweatherpredictioninHeipang airport- Jos Plateau, Nigeria where variables used forthe studywere temperature, rainfall and wind speed. The ARIMA modelwasalso studiedin [18]andacomparisonwasmadebetweenARIMA andtheArtificial NeuralNetwork(ANN) to predict rainfallin Mauritius.In [19]a comparativestudyof themostrecent computationalintelligence techniqueswhich havebeenappliedformeteorologicaltime seriespredictionpurposewasmade. AmodifiedversionoftheK-NNapproach wasappliedtopredictweather datasuchassolarradiation,maximumandminimumtemperature,andrainfallin[20].In [21]amodelforthepredictionofweather andsolarunitsbasedontheK-NNalgorithmwasused.In[22],K-NNwasused withanapproachofdataminingfortheimplementationofthepredictionsystem. In[20]thepredictionofdailyweather data for climate forecastsusing the non-parametric nearest-neighbor re-samplingtechnique was applied. A comparative studyofMovingAverageonRainfallTimeSeriesData forRainfallForecastingBasedonEvolvingNeuralNetworkClassifier wasutilizedin[23].Also,theArtificial NeuralNetworkwasappliedforweather forecastingin[24–26].In[27],prediction oftheDutchweather wasdone usingrecurrentneuralnetworks. In[28],theneuralnetwork trainingmodelwasusedfor weatherforecastingusingfireworksalgorithm.

Inthispaper,an IoT basedshort-term weather predictionarchitectureis implemented.Themain noveltyofthepaper isthe formulationofthree variantsoftheMultipleLinear Regression(MLR) algorithmwhichare obtainedexperimentally to determine thecombinationof parameters yielding the bestperformance. Moreover, adaptive versionsof theMLR and K-NN algorithms were developed by the use ofa time-varying trainingwindow. Finally, three adaptiveselection criteria wereintegratedinthesystemtoselectthemostappropriatepredictionalgorithmtobeusedforeachreal-timeforecast.The systemwasusedtocaptureweatherparameters suchastemperature,humidity,atmosphericpressure,rainfall,luminosity, wind-speed,andwinddirectionforaperiodofeightdaysonthecampusoftheUniversityofMauritius.Thebestadaptive schemesareabletopredicttheseparameterswithan overallpercentageerrorof6.58%ascomparedtonon-adaptiveones whichpredictedwithaworstcasepercentageerrorof13.59%.

The rest of the paper is organized as follows: Section 2 presents a detailed explanation of the system architecture.

Section 3 provides an overviewof forecastingalgorithms. Section 4 presentsthe results accompanied by comprehensive analysisandﬁnallytheworkisconcludedinSection5.

2. Systemarchitecture

ThissectiondescribestheHardwareSet-up,conﬁgurationofthemicro-controllersandthedatabaseserver.

2.1. Hardwareset-up

TheblockdiagramoftheproposedweatherforecastingsystemisshowninFig.1.

Theweatherforecastingsystemconsistsofoneweather-monitoringnode.Thenodeiscapableofmeasuringtemperature, atmosphericpressure,luminosity,humidity,winddirection,windspeedandrainthroughtheweather meterandthetem- peratureandhumiditysensor.Thesensorsareconnectedtotheweathershield,whichisinturnstackedontotheArduino micro-controller.TheXBeeShieldandWi-FimodulesallowtheArduinomicro-controllerstobeconnectedtotheRaspberry Pi3wirelessly.

TheRaspberryPi3isapowerfulmicrocontrollerwhichisusedtoaggregatethedatafromtheweather-monitoringnode and actsas a gateway to the local databaseserver. The Raspberry Pi 3 also functionsas an edge IoT device which can performvariousprocessingtasksatanearlystagesoastooﬄoadtheprocessingburdenfromtheapplicationserveralone.

AstraightforwardintegrationofaJavaprogramcanberunontheRaspberryPi3toperformdatanormalization,dataerror detection,missingdatadetectionandaggregationofdatafromseveralnodestransmittingdatainparallel.TheRaspberryPi 3isconnectedto thedatabaseserverthroughan InternetRouter.The applicationserver allowstheuserofthesystemto querythedatabaseorreceivealerts.Typically,theuserqueriesthedatabaseusingamobilephone.

The valuesrecordedby the sensorsofthe weather monitoringnode are temperature, humidity,atmosphericpressure, luminosity,windspeed,winddirection, andrainfall.Fig.2illustrateshowthesensorvaluesarerecordedandtransmitted.

ThesensorvaluesobtainedfromthenodeisstoredinJSONformatwhichisastandardwayofstoringvariablenamesand theircorresponding valuesinkey-valuepairs,andtransmittedthroughthewirelesslinktotheRaspberryPiforprocessing andstorage.

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Fig. 1. Block diagram of the proposed weather forecasting system.

Fig. 2. Sensor values recorded.

Fig. 3. XBee Shield setup with Arduino Uno and Weather Shield.

Fig.3showshowthe XBeeshieldhasbeen usedintheproject.The XBeeshieldis mountedon theArduinoUnoand the weather shield is mounted onto the latter. The Xbee Shield allows data from the weather shield and meters to be transmittedtotheRaspberryPi3usingWi-Fi.

TheXBeeandweather shieldsare mounteddirectlyonto theArduino.Since bothshields usepins 2and3bydefault;

theXBeeshieldusesthemforcommunicationwhereastheweathershieldusesthemtocollectthedatafromthewindand rainsensors.Therefore,are-wiringtoovercomethisoverlapwasrequired.Theunusedpins4and5onthemicro-controller wereusedfortheXBeeshieldtoperformthetransceiveroperationsandthedefaultpinswereusedwiththeweathershield.

There-wiringperformedisshowninFig.4.

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Fig. 4. Re-wiring between XBee Shield, Arduino Uno and Weather Shield.

2.2. Micro-controllerconﬁgurationandprogramming

TheprocessesinvolvedintheconﬁgurationsoftheArduinomicro-controllerandtheRaspberryPi-3areexplainedinthe followingsub-sections.

2.2.1. Arduinoconﬁgurationandprogramming

TheArduinoisprogrammedusingtheC/C++programminglanguagewherebyconﬁgurationshavebeenmadeto:

- Readdatafromthedifferentweathersensors,

- CombineallsensorvariablesandcorrespondingsensorvaluesinaJSONstring,

- TransmitthesensordatainJSONformatthroughtheXBeeWi-Fimoduleonaspeciﬁcsocketconnectiontobereceived bytheRaspberryPi-3.

2.2.2. RaspberryPi-3conﬁgurationandprogramming

TheRaspberry Pi-3hasaLinux-basedoperatingsystem(RaspbianJessy)mountedonit andfilesare amendedto con- figurethedeviceasawireless router.Javaprogramsarealsorunfortheinterceptionofthedataonthesocketconnection throughthewirelessinterfaceandinsertingittoaspecifictableinadatabaseserverthroughtheEthernetinterface.

2.3. Webserverarchitectureandfordatastorageandacquisition

The MySQLdatabasewhichcomes alongthe XAMPPsoftwareisused extensivelyinthiswork forthelocalstorageof recordedsensordata.ThesevaluesarethenaccessedbytheJavaapplicationforperformingpredictiveanalyticsandcompute forecastedweatherdata.Thelatteristhen insertedintoatableinthesameMySQLdatabase.The predictedvaluescanbe queriedusinganapplicationrunningonamobilephone.ThewholeprocessworksisasshowninFig.5.

2.3.1. Databasearchitecture

Twotablesareimplementedinasingledatabasefortheproposed system.Onetableisusedforstoringtheaggregated sensorvaluesandthesecondoneisusedtosavepredictedvalueswhichareaccessedbythemobileapplication.

The fields for“id” and “time” are automatically generated andadded when newsets of sensorvalues are written to the table in the database. The “time” field refers to the timestamp atwhich the data wascollectedand inserted inthe database.Thefields“windGustmph”,“windGustdir”,“humidity”,“temperature”,“rainIn”,“pressure”,and“lightLevel” referto the sensorvaluesof wind speed,wind direction, humidity,temperature, rain level,atmosphericpressure, andluminosity respectively.AsampleofthevaluesrecordedinthetableisasdepictedinFig.6.

The MySQL databaseused forthis work is a low complexitydatabase asit involves simpler conﬁgurations or setup.

However,anupgradetoNoSQLdatabases(CassandraorMongoDB) wouldbemoreappropriatewhendeployingthesystem onalargerscale.TheadvantageofNoSQLinthiscasewouldbepresentedduetolargevolumesofdatabeingdealtwith.

Datafromsensorsaresentinreal-timetothedatabase.Thesystemwillobtainonlytheparametersfortheforecasting algorithmanddataforawindowcorrespondingtothelast120mintoperformtheforecasting.Assuch,theburdenonthe systemisconstantanddeterminedbythewindowsizesetandnotallthedatafoundinthedatabase.Hence,itwillnotbe anissuetodeployinlargerareas.Moreover,tomovetoawiderarea,wewillneedadditionalsensornodestoofferawider coverage.

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Fig. 5. Architecture for data acquisition and query from mobile device.

Fig. 6. Sample of Recorded Sensor values.

3. Forecastingalgorithmsandsoftwaredesign

Thissectiongivesanoverviewoftheforecastingalgorithmsthatwereinvestigatedinthiswork.

3.1. Forecastingalgorithms

Inthissectionthefollowinggeneralsymbolshavebeenusedtodescribethedifferentvariablesintheequations:

T temperatureindegreesCelcius

H %Humidity

L Lightintensityin R Rainfallinmm

P AtmosphericpressureinPa.

WD Winddirectionconvertedtoadegreesscale.

WS Windspeedinm/s

W Thetrainingwindowwhichwasﬁxedto120minsinthiswork.

t Thetimeindex.

3.1.1. Multiplelinearregression

Inthiswork,withmultiple linearregression,equationsinvolvingageneralparameter P1 tobe predictedattimet+1 withrespecttothepreviouslyrecordedvalueofP1attimetandanotherrelatedparameterPxwereformulatedasperthe followinggenericcombination:

P1t+1=a0

(

^W

)

⁺^a¹

(

^W

)

^P¹^t⁺^a²

(

^W

)

^P^x^t ⁽¹⁾

where,

PxtdenotesanyotherrelatedparameterthatcouldinﬂuencethevalueofP1t.

a₀(W),a₁(W)anda₂(W)arecoeﬃcientsdeterminedforatrainingwindowofsizeWwhichwaskeptat120min.

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Thesecoeﬃcientswererecomputedafterevery120mintoadjustthemodeltothenewlyrecordedvaluesobtainedfrom theweathersensors.

Three different combinations of multiple linear regression equations denoted as MLR, MLR1 and MLR2 were derived experimentallyforthesevenweatherparameters.Thesecombinationsaredescribedasfollows:

MLR. Withthismodel,thefollowingcombinationofequationswereused:

T_t+1=a₀

(

^W

)

⁺^a1

(

^W

)

^T^t⁺^a2

(

^W

)

^H^t ⁽²⁾

InEq.(2),thevalueofthe temperatureforthenext timeinterval, i.e.T_t₊₁ ispredictedfromtheprevious temperature value Tt andprevious humidity valueHt.The constantsa₀(W), a₁(W),anda₂(W) areobtainedfromthe regressionmodel foratrainingwindowofsizeW.Thedetailsoftheequations(3)–22)followthesameconventionasforEq.(2)withtheir respectiveparameters.

Ht+1=a0

(

^W

)

⁺^a¹

(

^W

)

^H^t⁺^a²

(

^W

)

^T^t ⁽³⁾ L_t+1=a0

(

^W

)

+a1

(

^W

)

^L^t+a2

(

^W

)

^T^t ⁽⁴⁾ Pt+1=a0

(

^W

)

⁺^a¹

(

^W

)

^P^t⁺^a²

(

^W

)

^T^t ⁽⁵⁾ Rt+1=a0

(

^W

)

⁺^a¹

(

^W

)

^R^t⁺^a²

(

^W

)

^T^t ⁽⁶⁾ WS_t+1=a₀

(

^W

)

⁺^a1

(

^W

)

^W^S^t⁺^a2

(

^W

)

^T^t ⁽⁷⁾

WDt+1=a0

(

^W

)

⁺^a¹

(

^W

)

^W^D^t⁺^a²

(

^W

)

^T^t ⁽⁸⁾

MLR1.

Tt+1=a0

(

^W

)

⁺^a¹

(

^W

)

^T^t⁺^a²

(

^W

)

^L^t ⁽⁹⁾ Ht+1=a0

(

^W

)

⁺^a¹

(

^W

)

^H^t⁺^a²

(

^W

)

^L^t ⁽¹⁰⁾ Lt+1=a0

(

^W

)

⁺^a¹

(

^W

)

^L^t⁺^a²

(

^W

)

^H^t ⁽¹¹⁾ P_t+1=a₀

(

^W

)

+a₁

(

^W

)

^P^t+a₂

(

^W

)

^H^t ⁽¹²⁾ Rt+1=a0

(

^W

)

⁺^a¹

(

^W

)

^R^t⁺^a²

(

^W

)

^H^t ⁽¹³⁾ WSt+1=a0

(

^W

)

⁺^a¹

(

^W

)

^W^S^t⁺^a²

(

^W

)

^H^t ⁽¹⁴⁾

WD_t+1=a₀

(

^W

)

⁺^a1

(

^W

)

^W^D^t⁺^a2

(

^W

)

^H^t ⁽¹⁵⁾

MLR2.

Tt+1=a0

(

^W

)

⁺^a¹

(

^W

)

^T^t⁺^a²

(

^W

)

^P^t ⁽¹⁶⁾ Ht+1=a0

(

^W

)

⁺^a¹

(

^W

)

^H^t⁺^a²

(

^W

)

^P^t ⁽¹⁷⁾ Lt+1=a0

(

^W

)

⁺^a¹

(

^W

)

^L^t⁺^a²

(

^W

)

^P^t ⁽¹⁸⁾ Pt+1=a0

(

^W

)

⁺^a¹

(

^W

)

^P^t⁺^a²

(

^W

)

^L^t ⁽¹⁹⁾ R_t+1=a₀

(

^W

)

+a₁

(

^W

)

^R^t+a₂

(

^W

)

^L^t ⁽²⁰⁾ WSt+1=a0

(

^W

)

⁺^a¹

(

^W

)

^W^S^t⁺^a²

(

^W

)

^L^t ⁽²¹⁾ WDt+1=a0

(

^W

)

⁺^a¹

(

^W

)

^W^D^t⁺^a²

(

^W

)

^L^t ⁽²²⁾

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3.1.2. Autoregressiveintegratedmovingaverage(ARIMA)[29]

ARIMAisastochastictimeseriesmodelﬁrstpopularizedbyBoxandJenkins[30].AnARIMAmodelisageneralizationof anAutoregressiveMovingAverage(ARMA)modelthatisusedtopredictavalueintimeseriesdataasalinearcombination ofitspastvaluesandpasterrors.Themodelisusedontimeseriesdatawhichcanbemadestationarybydifferencing[31]. ThegeneralexpressionofARIMAforecastingisgivenbelow:

Yi+1=

μ

+

φ

1Yi−1+· · · +

φ

pYi−p+

θ

1ei−1+· · · +

θ

qei−q (23) where,

• prepresentstheorderofauto-regression

• qistheorderofmovingaverage(numberofpasterrorterms)

•

φ

^is^the^slope^coeﬃcient

•

θ

îs^the^movingâverage^coefficient

• eistheerrorterm

•

μ

^is^a^constant^term

3.1.3. K-nearestneighbors(K-NN)[29]

TheK-NNalgorithmdependsonlyonthegivendatasetandauser-deﬁnedconstantparameternamelyK.Thefollowing proceduresexplaintheK-NNalgorithm:

1.Withadistancefunction,theKreadingsnearesttothepredictionpoint(nextinterval)isretrieved.

2.TheforecastedoutputiscomputedasthemeanoftheKnearestreadingsgiveninthefollowingequation:

Y_i₊₁= 1 K

K

i=1

y_i (24)

where,

• Kistheneighborhoodsize.

• y_iisthenearestreading.

3.1.4. AdaptivemultiplelinearregressionandK-NN

InordertomaketheMLR, MLR1andMLR2algorithmsadaptive,thetrainingwindowW,usedtocomputetheir coeffi- cientsa₀,a₁ anda₂shouldbeupdatedeachtimeanewparameterisreadfromthesensors.Inotherwordseverytseconds anothersetofcoefficientsarecomputedfortheMLRmodel.Themainmodificationisthatthecoefficientsarenowafunc- tionofWt i.e.a trainingwindow ofsizeW whichischanged everyt secondsinsteadofa fixed trainingwindow W.The modifiedequationsareasfollows:

AdaptiveMLR. Withthismodel,thefollowingcombinationofequationswereused:

Tt+1=a0

(

^W^t

)

⁺^a¹

(

^W^t

)

^T^t⁺^a²

(

^W^t

)

^H^t ⁽²⁵⁾

Ht+1=a0

(

^W^t

)

⁺^a¹

(

^W^t

)

^H^t⁺^a²

(

^W^t

)

^T^t ⁽²⁶⁾

Lt+1=a0

(

^W^t

)

⁺^a¹

(

^W^t

)

^L^t⁺^a²

(

^W^t

)

^T^t ⁽²⁷⁾

Pt+1=a0

(

^W^t

)

⁺^a¹

(

^W^t

)

^P^t⁺^a²

(

^W^t

)

^T^t ⁽²⁸⁾

Rt+1=a0

(

^W^t

)

⁺^a¹

(

^W^t

)

^R^t⁺^a²

(

^W^t

)

^T^t ⁽²⁹⁾

WSt+1=a0

(

^W^t

)

⁺^a¹

(

^W^t

)

^W^S^t⁺^a²

(

^W^t

)

^T^t ⁽³⁰⁾

WDt+1=a0

(

^W^t

)

⁺^a¹

(

^W^t

)

^W^D^t⁺^a²

(

^W^t

)

^T^t ⁽³¹⁾

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AdaptiveMLR1

Tt+1=a0

(

^W^t

)

⁺^a¹

(

^W^t

)

^T^t⁺^a²

(

^W^t

)

^L^t ⁽³²⁾ Ht+1=a0

(

^W^t

)

⁺^a¹

(

^W^t

)

^H^t⁺^a²

(

^W^t

)

^L^t ⁽³³⁾ Lt+1=a0

(

^W^t

)

⁺^a¹

(

^W^t

)

^L^t⁺^a²

(

^W^t

)

^H^t ⁽³⁴⁾ P_t+1=a₀

(

^W^t

)

⁺^a1

(

^W^t

)

^P^t⁺^a2

(

^W^t

)

^H^t ⁽³⁵⁾ R_t+1=a₀

(

^W^t

)

+a₁

(

^W^t

)

^R^t+a₂

(

^W^t

)

^H^t ⁽³⁶⁾ WS_t+1=a₀

(

^W^t

)

+a₁

(

^W^t

)

^W^S^t+a₂

(

^W^t

)

^H^t ⁽³⁸⁾

WDt+1=a0

(

^Wt

)

+a1

(

^Wt

)

^W^Dt+a2

(

^Wt

)

^Ht (39) AdaptiveMLR2

T_t+1=a₀

(

^W^t

)

⁺^a1

(

^W^t

)

^T^t⁺^a2

(

^W^t

)

^P^t ⁽⁴⁰⁾ H_t+1=a₀

(

^W^t

)

+a₁

(

^W^t

)

^H^t+a₂

(

^W^t

)

^P^t ⁽⁴¹⁾ L_t+1=a₀

(

^W^t

)

+a₁

(

^W^t

)

^L^t+a₂

(

^W^t

)

^P^t ⁽⁴²⁾ P_t+1=a0

(

^Wt

)

+a1

(

^Wt

)

^Pt+a2

(

^Wt

)

^Lt (43) Rt+1=a0

(

^W^t

)

⁺^a¹

(

^W^t

)

^R^t⁺^a²

(

^W^t

)

^L^t ⁽⁴⁴⁾ WSt+1=a0

(

^W^t

)

⁺^a¹

(

^W^t

)

^W^S^t⁺^a²

(

^W^t

)

^L^t ⁽⁴⁵⁾

WDt+1=a0

(

^W^t

)

⁺^a¹

(

^W^t

)

^W^D^t⁺^a²

(

^W^t

)

^L^t ⁽⁴⁶⁾

Adaptive K-NN. Foradaptive K-NNalsothe trainingwindow ischanged every t secondsandthe modiﬁedequation is as follows:

Yt+1= 1 K

K t=1

Yt f orK

^W^t ⁽⁴⁷⁾

3.1.5. Proposedadaptiveselectionalgorithms

Threeadaptive selectionalgorithms,denoted asA1,A2andA3respectivelyhavebeendeveloped.Therationale behind theseadaptivealgorithmsistoselectthebestpredictionalgorithmforpredictingagivenparameterbasedonthepredictor thatachievedthebestperformanceforthepreviousprediction.Thefollowingselectedpredictionalgorithmswereincluded inthesetofpredictionalgorithmstobeusedfortheadaptiveselectionalgorithms:

1. ARIMA.

2. AdaptiveK-NN.

3. AdaptiveMultiLinearregression.

4. AdaptiveMultiLinearregression1.

5. AdaptiveMultiLinearregression2.

LetP^x denoteanyoneoftheabovepredictorswherex=1,2,3,4,5inthiscaseandP¹representsArima,P² K-NNandso on.

ConsiderFig.7andnotationsthatwillbeusedinformulatingthepredictionalgorithms:

Theparametersintheaboveﬁgurearedeﬁnedasfollows:

T_A Theactualorcurrenttime.

T_S Thetimeatwhichtheﬁrstvalueinthetrainingwindowwasrecorded.

TM Themid-timeintervalbetweenTS andTA.

(10)

Fig. 7. Timing diagram for Algorithm A1.

Fig. 8. Timing diagram for Algorithm A2.

T_A₊_I Thetime atwhichthevaluesoftheparameteristobepredicted.ThesymbolIrepresentsanintervalinminutes aftertheactualtimeT_A.

W Thetrainingwindowsizeinminuteswhichessentiallyrepresentasetofpre-observedvaluesoftheparameterto bepredictedoverthepastWminutesgoingbackfromtheactualtimeT_A toT_S.

V_A TheactualorcurrentvaluesoftheparameterbeingmeasuredattimeT_A. V_S ThevalueoftheparameterbeingmeasuredattimeTs.

V_M ThevalueoftheparameterbeingmeasuredattimeT_M. Vˆ_A^x₊_I ThevalueoftheparameterpredictedwithP^x attimeT_A₊_I.

AlgorithmA1isformulatedasfollows:

1.UsingthetrainingsetvaluesrecordedfromtimeT_StoT_A₋₁ i.e.thevaluesV_S toV_A₋₁,predictthevalueattimeT_Awith eachpredictorP^x asfollows:

Vˆ_A^x=P^x

(

^V^S^,^V^A−1

)

⁽⁴⁸⁾

where,Vˆ_A^x isthe value ofthe parameter predictedattimeT_A withagivenpredictorP^xthatusesvaluesfromV_S toV_A₋₁ toperformtheprediction.

Moreexplicitlyforthefivepredictorsusedinthisspecificcase,therewillbefivepredictionsobtainedforthevaluesat timeT_Aasfollows:

Vˆ_A¹=P¹

(

^V^S^,^V^A−1

)

^; ^V^ˆA²=P²

(

^V^S^,^V^A−1

)

^;^V^ˆA³=P³

(

^V^S^,^V^A−1

)

^; ^V^ˆA⁴=P⁴

(

^V^S^,^V^A−1

)

^; ^V^ˆA⁵=P⁵

(

^V^S^,^V^A−1

)

⁽⁵⁰⁾

2.ComputethesquaredEuclideandistance,betweentheactualvalue oftheparameterVA observedtime TA andthepre- dictedvalueVˆ_A^x obtainedwitheachpredictorP^xasfollows:

SE_A^x=

VA−Vˆ_A^x

2

(51) InthiscasetherewillbeﬁvevaluesoftheMSEobtainedasfollows:

SE_A¹=

VA−Vˆ_A¹

²

; SE_A²=

VA−Vˆ_A²

²

; SE_A³=

VA−Vˆ_A³

²

; SE_A⁴=

VA−Vˆ_A⁴

²

; SE_A⁵=

VA−Vˆ_A⁵

² ₍₅₂₎

3.PredictthevalueattimeT_A₊_Iwiththepredictor,P^x,givingthelowestMSEforthepredictedvalueattimeT_A: Vˆ_A^x₊₁=P^x

(

^VS,V_A

)

x=Index

min[SE^x_A]

=Index

min

SE_A¹,SE_A²,SE_A³,SE_A⁴,SE_A⁵

(53)

ConsiderFig.8whichisthetimingdiagramforalgorithmA2.

Thefollowingadditionaltermsaredeﬁnedfortheabovediagram:

(11)

W/2:ItisequivalenttohalfthewindowsizeWandcontainsvaluesoftheparameterrecordedfromtime T_Mto T_A i.e.

thevaluesV_MtoV_A. V

x

M isavaluepredictedattimeT_MwithpredictorP^x usingavailablevaluesfromV_StoV_M₋₁. Vˆ_M^x₊₁ isavaluepredictedattimeT_M₊₁withpredictorP^x usingavailablevaluesfromV_StoV_M.

Vˆ_M^x₊_iisavaluepredictedattimeT_M₊_iwithpredictorP^xusingavailablevaluesfromV_StoV_M₊_i₋₁.Theindexitakesvalues from1toW/2.

SE_M^x,SE^x_M₊₁,SE_M^x₊₂,...SE^x_M₊_i...SE_A^x represent the squared Euclidean distances between the actual values V_M,V_M₊₁,V_M₊₂,….V_M₊_i…..V_A andV^x_M,V^x_M₊₁,V^x_M₊₂,...V^x_M₊_i...V^x_A respectively.

Thealgorithmproceedsasfollows:

1. Predictthevaluesfromtime T_M toT_A i.e.V

x M,V

x M+1,V

x M+2,...V

x M+i...V

x

A ataoneminuteintervalswitheachpredictorP^x asfollows:

Vˆ_M^x =P^x

(

^VS,V_M−1

)

⁽⁵⁴⁾

Moreexplicitly:

Vˆ_M¹=P¹

(

^V^S^,^V^M−1

)

^; ^V^ˆM²=P²

(

^V^S^,^V^M−1

)

^; ^V^ˆM³=P³

(

^V^S^,^V^M−1

)

^;^V^ˆA⁴=P⁴

(

^V^S^,^V^M−1

)

^;^V^ˆA⁵=P⁵

(

^V^S^,^V^M−1

)

⁽⁵⁵⁾

Ingeneral:

Vˆ_M+^x _i=P^x(^VS,V_M+_i₋₁)^and:

Vˆ_M¹₊_i=P¹

(

^VS,V_M₊_i₋₁

)

; Vˆ_M²₊_i=P²

(

^VS,V_M₊_i₋₁

)

; Vˆ_M³₊_i=P³

(

^VS,V_M₊_i₋₁

)

;

Vˆ_M⁴₊_i=P⁴

(

^VS,VM+i−1

)

; Vˆ_M⁵₊_i=P⁵

(

^VS,VM+i−1

)

⁽⁵⁶⁾

2. Compute the squared Euclidean distances SE_M^x,SE^x_M₊₁,SE_M^x₊₂,...,SE_M^x₊_i,...,SE_A^x between the actual values V_M,V_M₊₁,V_M₊₂,….V_M₊_i…..V_A andVˆ_M^x,Vˆ_M^x₊₁,Vˆ_M^x₊₂,..., Vˆ_M^x₊_i,...,Vˆ_A^x respectivelyforeachalgorithmasfollows:

SE_M^x =

VM−Vˆ_M^x

2

(57) Moreexplicitly:

SE_M¹ =

VM−Vˆ_M¹

2

; SE_M² =

VM−Vˆ_M²

2

; SE_M³ =

VM−Vˆ_M³

2

; SE_M⁴ =

VM−Vˆ_M⁴

2

; SE_M⁵ =

VM−Vˆ_M⁵

2

(58) Ingeneral:

SE_M^x₊_i=(^VM+i−Vˆ_M^x₊_i)²and more explicitly:

SE_M+¹ _i=

VM+i−Vˆ_M+¹ _i

2

; SE²_M+_i=

VM+i−Vˆ_M+² _i

2

; SE³_M₊_i=

VM+i−Vˆ_M³₊_i

2

; SE_M+⁴ _i=

VM+i−Vˆ_M+⁴ _i

²

; SE⁵_M+_i=

VM+i−Vˆ_M+⁵ _i

²

(59) 3. Compute the mean squared error for each predictor for all predictions made from TM to TA of the values

Vˆ_M^x,Vˆ_M^x₊₁,Vˆ_M^x₊₂, ..., Vˆ_M^x₊_i, ..., Vˆ_A^x.ForagivenpredictorP^xtakingifrom1toW/2,weobtaintheMSEasfollows:

MSE^x= 2 W

W/2

i=1

SE_M^x₊₁₋_i (60)

Moreexplicitly:

(

^MSE

)

¹= 2 W

(^W/2) (ⁱ=1)

SE₍¹_M₊₁₋_i₎;

(

^MSE

)

²= 2 W

(^W/2) (ⁱ=1)

SE²₍_M₊₁₋_i₎;

(

^MSE

)

³= 2 W

(^W/2) (ⁱ=1)

SE₍³_M₊₁₋_i₎;

(

^MSE

)

⁴⁼ _W² ⁽

W/2) (i=1)

SE₍⁴_M+1_−i₎;

(

^MSE

)

⁵⁼_W² ⁽

W/2) (i=1)

SE⁵₍_M+1_−i₎ (62)

4. PredictthevalueattimeT_A₊_Iwiththepredictor,P^x,givingthelowestMSEforthepredictedvaluesbetweenT_MandT_A: Vˆ_A^x₊₁=P^x

(

^V^S^,^V^A

)

x=Index

min

MSE^x

=Index

min

MSE¹,MSE²,MSE³,MSE⁴,MSE^x

₍₆₃₎