ITRON S FORECASTING BROWN BAG SEMINAR Usng Condtonal Heteroskedastc Varance Models n Load Research Sample Desgn Dr. J. Stuart McMenamn March 6, 2012
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Motvaton for the Presentaton» Itron s addng a sample desgn component to the Itron Load Research System (LRS). Varance Model Estmaton Sze Breakpont analyss Stratfcaton analyss for populaton data Optmal sample sze and precson calculatons Sample pont selecton» One of the advanced approaches that s ncluded s called Model-Based sample desgn.» Purpose of ths presentaton s to share what we have learned about model based approaches and to relate the approaches to varance modelng (CH, ARCH, GARCH) 3
Typcal Desgn Problems» Class Load Research Desgn a sample to estmate class loads at the tme of system peak For each class, stratfy the populaton, and select ponts Select enough ponts to estmate class loads +- 10% at tme of peak Typcal stratfcaton: - Sze (annual sales, seasonal sales, or peak month sales) - End use (wth electrc heat vs. wthout electrc heat) - Segment (housng type, buldng type, ndustry) - Geography (jursdcton)» Program Evaluaton Desgn a sample of partcpants p to estmate program mpacts Stratfy the partcpant populaton by some measure of sze - Sales (annual or seasonal) - Expected mpacts, ncentve payment level, 4
Load Research Sample Expanson Each blue lne represents the hourly loads of one sample customer n the General Servce Small stratum. The black lne s the stratum average. The goal of sample expanson s to estmate the hourly load and confdence bands for the class. 5
Sample Desgn Questons» How bg does the sample need to be to reach a target precson level.» Or Wth a sample sze of n, how do we get the best precson.» To answer the questons about precson and sample sze, you need to know somethng about the thng you are tryng to estmate: An expected average value (y average) A rato to somethng that s known (y/x rato) A measure of dsperson n the populaton (y standard devaton)» Model based desgn can help answer key questons. Uses are: Defne stratfcaton level and sze breakponts Allocate sample ponts to strata Provde desgn parameters for precson calculatons 6
Statstcs for Mean Per Unt (MPU) Estmaton n = 66 Average yavg s n y Standard Devaton ystddev s y n 11 yavg 2 Coeffcent of Varaton CV s a untless measure of varablty ycv ystddev yavg 7
MPU Precson Calculatons n = 66 Standard error of the estmated average value (yavg) AvgStdErr ystddev SdD n fpc 95% Confdence Band Fnte populaton correcton factor s usually close to 1.0. fpc s 1 n N s s Relatve precson s the wdth of the confdence band relatve to the estmated value. All we need to know s the CV and the sample sze (n) RP 1.96 AvgStdErr yavg 1.96 ycv n 8
Precson vs. Sample Sze Relatve Precson Calculate the precson for 1.96 AvgStdErr ycv RP 1.96 a gven sample sze n: yavg n Example: ycv=.522, n=66 RP = 13% Requred Sample Sze 1.96 ycv n RP 2 Calculate the sample sze requred for a target precson at a gven confdence level Example: ycv=.522, RP = 10% n=105 It gets more complex wth stratfcaton, but not much. Formulas are the same wthn each stratum. Weght across the strata to get overall results. 9
Mean per Unt (MPU) Desgn Statstcs ycv.52 ycv.55 ycv.40 xcv.42 xcv.30 xcv.12 10
Rato Methods and Auxlary Varables» Auxlary varable (X) -- somethng that s known for all customers: Annual energy Seasonal energy Maxmum demand for demand metered classes» Rato methods can mprove precson. Ths works well when the target varable and the auxlary varable are strongly correlated.» There are two types of rato expanson: Separate rato (SR) uses monthly energy values for each stratum. Combned rato (CR) uses monthly energy values for the class. Typcally CR s used because populaton energy s avalable at the class level but not at the stratum level.» Regardless of the varable used n sample desgn, the varable used n expansons s typcally revenue month or calendar month energy. 11
Rato Estmaton wth Auxlary Varable Y (KWh) Rato (KWh/MWh) R s s y n x n X (MWh) Correcton Factor k Pop s x N x n ErrRato s a untless measure of y varablty relatve to the rato lne Error Rato ErrStdDev s y R x n 1 2 ErrStdDev ErrRato yavg 12
Rato Precson Standard Devaton of Resduals ErrStdDev s y R x n 11 2 95% Confdence Band Standard Error wth Rato Estmaton ErrStdDev AvgStdErr fpc n Relatve Precson s the wdth of the confdence band relatve to the estmated value. All we need s the Error Rato and the sample sze (n) RP 1.96 AvgStdErr k yavg ErrRato 1.96 n 13
MPU Versus Rato Rato Estmaton Mean Per Unt Estmaton RP 1.96 ycv n RP ErrRato 1.96 n Generally, f the correlaton between X and Y s strong (>.5), rato estmaton wll provde better precson than mean per unt estmaton. 14
Combned Rato Method Stratfed ycv.52 ycv.55 ycv.40 Corr.62 Corr.71 Corr.35 ErrRato.46 ErrRato.40 ErrRato.42 Combned rato estmates use one rato lne for all the strata. Resduals are measured around ths lne. Combned Rato =.269 15
CH Models» Structural Model: e X B Y» Constant Varance (Homoskedastc) v e Var» Condtonal Heteroskedastc (CH) Z f v e Var» Autoregressve Condtonal Heteroskedastc (ARCH(1)) 2 1 t 1 0 t t e a a v ) (e Var» Generalzed Autoregressve Condtonal Heteroskedastc (GARCH(1,1)) 1 t 1 2 1 t 1 0 t t v b e a a v ) (e Var 16
Model Based Sample Desgn» Model based sample desgn utlzes varance models to develop desgn parameters for sample desgn efforts. Ths approach was frst appled to load research by Roget Wrght n the late 1980 s.» Model based desgn uses a condtonal heteroskedastc (CH) model of the error varance n the Y drecton.» Ths s smlar to ARCH and GARCH models that are used to model volatlty n tme seres problems. The same nonlnear estmaton methods work for CH models.» Of course, ths requres data from a pror sample to estmate the model parameters. 17
CH Model Used n Model Based Sample Desgn» Y v R X s 2 e where s k X a a=.50 k=.64» a = elastcty of the error standard devaton (s) wth respect to X» R s set to.2 KW/MWh n all cases a=1.0 k=.08 a=0.0 k=4.8 18
Estmatng Model Parameters» Varance Model: Y v R X s 2 e where s k x a» Parameters are R, a, k» Iteratve e estmaton methods -- Assume a, k. Estmate R usng weghted least squares -- Use R to compute e. Estmate a from -- Repeat untl convergence» Maxmum Lkelhood -- Estmates R, a, k smultaneously Ln 2 e Lnk a LnX -- Provdes a consstent set of estmated parameter standard errors -- Requres nonlnear optmzaton 19
Varance Model for Resdental Non Heatng 20
Varance Model for Resdental Heatng 21
Varance Model for General Servce (GS) 22
Varance Model for Large General Svc (LGS) 23
Varance Models and Sze Stratfcaton» Sze stratfcaton steps: Defne sze bns for frequency analyss Process populaton nto bns Compute allocaton measure - For Dalenus-Hodges (DH), compute Sqrt(Freq) q) - For Model Based, compute Sum (S) = Sum (kx a ) - For volume based allocaton, compute Sum (X) - For proportonal p allocaton, compute Frequency Compute cumulatve sum of the allocaton measure Break cumulatve dstrbuton nto equal amounts» MPU desgns often use DH rule» Rato desgns often use Model Based rule 24
Dalenus-Hodges Breakponts (GS Class) Customer Frequency by Bn Cumulatve Sqrt (Frequency) 356.5 177.6 71.7 25
Model-Based Breakponts Customer Frequency by Bn Cumulatve Sum (kx a ) 348.5 ErrRato =.4, Elastcty =.75 176.66 86.0 26
Varance Models and Populaton Statstcs» Populaton Analyss steps Defne stratfcaton rules Flter and sort populaton data Compute populaton statstcs by strata - Number of customers (N), Average auxlary varable values (X) Compute model based Error Ratos based on: - Overall Error Rato - Elastcty parameter (a) 27
Varance Models and Sample Desgn» Sample Desgn Steps Specfy desgn approach (MPU or Rato) Enter populaton statstcs (N, X) Select allocaton approach - Neyman optmal allocaton (allocate proportonal to N ystddev) - Model based (allocate proportonal to model sgma) Enter desgn assumptons - For MPU: Y Standard Devaton or CV - For Model Based: Model parameters (ErrRato, Elastcty) Select desgn approach - Specfy sample sze, allocate for optmal precson - Specfy target precson, solve for sample sze 28
MPU Desgn 29
Model Based Desgn 30
Comparson of MPU and CR Results 3500 3000 2500 Mean Per Unt Combned Rato 2000 1500 1000 500 0 140 120 100 Total Class KW 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Mean Per Unt Combned Rato Rato methods wll be more precse than MPU when the correlatons between KW and KWh are strong 80 60 40 20 0 Standard Error of Totals 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 31
Summary» We covered a lot Sample expanson concepts and formulas CH and other varance models Applcaton of CH models to sample desgn - Support sze breakpont analyss - Develop populaton statstcs t t and error ratos - Support sample optmzaton and precson calculatons» Model data from past studes (y and x) values can be used to estmate model parameters.» Model parameters can be used to desgn future samples 32
2012 Brown Bag Semnars» Usng Condtonal Heteroskedastc Varance Models n Load Research Sample Desgn - March 6, 2012» Commercal Sales Modelng and Forecastng June 19, 2012» Forecast Accuracy vs. Forecast Stablty September 18, 2012» Smart Grd Forecastng December 11, 2012 All at noon, Pacfc Tme All are recorded and avalable for revew after the sesson. 33
Questons? Press *6 2012 HANDS-ON WORKSHOPS to ask a» Forecastng Workshop February 8-9, Sydney queston» Forecastng 101 - Aprl 18-20, Orlando» Forecastng Workshop Aprl 19-20, Brussels» One-Day Modelng Workshop Introducton to SAE May 9, Las Vegas» One-Day Modelng Workshop Forecastng Through Tme May 9, Las Vegas» Fundamentals of MetrxND June 11-12, Boston» Fundamentals of Sales & Demand Forecastng September 20-21, 21 Boston» Fundamentals of Short-Term and Hourly Forecastng October 11-12, San Dego» Forecastng 101 November 7-9, San Dego OTHER FORECASTING MEETINGS» Australan Users Meetng February 10, Sydney» European Users Meetng Aprl 25-27, Prague» Energy Forecastng Week May 7-11, Las Vegas Annual ISO/RTO Forecastng Summt May 7-8 Long-Term Forecastng/EFG Meetng May 10-11» Itron Users' Conference October 21-23, San Antono For more nformaton and regstraton: www.tron.com/forecastngworkshops Contact us at: 1.800.755.9585, 1.858.724.2620 or forecastng@tron.com 34