Nonstationary time series forecasting and functional clustering using wavelets Application to electricity demand Jean-Michel POGGI Univ. Paris Sud, Lab. Maths. Orsay (LMO), France and Univ. Paris Descartes, France bENBIS Energy Demand Forecasting workshop Leuven, January 18, 2018 Joint works with Anestis ANTONIADIS (Univ. Grenoble, France & Univ. Cape Town, South Africa) Xavier BROSSAT (EDF R&D, France) Jairo CUGLIARI (Univ. Lyon 2, France) Yannig GOUDE (EDF R&D, France and Univ. Paris-Sud, Orsay, France)
Industrial motivation Wavelets to describe Functional Data The Kernel+Wavelet+Functional (KWF) model How to cope with non stationarities Clustering functional data Clustering individual data Outline Industrial motivation 1 The Kernel+Wavelet+Functional (KWF) model 2 How to cope with non stationarities 3 4 Clustering functional data Scale-oriented feature extraction Wavelet coherence based dissimilarity Clustering individual data 5 Jean Michel Poggi Nonstationary time series forecasting and functional clustering using wavelets
Industrial motivation Wavelets to describe Functional Data Electricity demand data The Kernel+Wavelet+Functional (KWF) model Electricity demand forecasting How to cope with non stationarities Aim Clustering functional data Functional time series Clustering individual data Outline Industrial motivation 1 The Kernel+Wavelet+Functional (KWF) model 2 How to cope with non stationarities 3 4 Clustering functional data Scale-oriented feature extraction Wavelet coherence based dissimilarity Clustering individual data 5 Jean Michel Poggi Nonstationary time series forecasting and functional clustering using wavelets
Industrial motivation Wavelets to describe Functional Data Electricity demand data The Kernel+Wavelet+Functional (KWF) model Electricity demand forecasting How to cope with non stationarities Aim Clustering functional data Functional time series Clustering individual data Electricity demand data (a) Long term trend (b) Annual cycle (c) Weekly pattern (d) Daily pattern Jean Michel Poggi Nonstationary time series forecasting and functional clustering using wavelets
Industrial motivation Wavelets to describe Functional Data Electricity demand data The Kernel+Wavelet+Functional (KWF) model Electricity demand forecasting How to cope with non stationarities Aim Clustering functional data Functional time series Clustering individual data Electricity demand forecasting (general) Short-term electricity demand: a classical model Y t + 1 = F ( Y t , Y t − k , . . . , Y t − K ; X t , X t − k ′ , . . . , X t − K ′ ; C t ) + ǫ ( t ) endogenous variables (instantaneous and lagged values of Y ) exogenous: meteorology ( X ) and calendar effects ( C ) As consumption habits depend hardly on the hour of the day, very often one model per instant is fitted. Data are measured every 30 minutes + forecasting the next day curve 48 models corresponding to each of the sampled instants of the day Jean Michel Poggi Nonstationary time series forecasting and functional clustering using wavelets
Industrial motivation Wavelets to describe Functional Data Electricity demand data The Kernel+Wavelet+Functional (KWF) model Electricity demand forecasting How to cope with non stationarities Aim Clustering functional data Functional time series Clustering individual data Electricity demand forecasting (general) Short-term electricity demand: an additive model Y t = � K k = 1 f 0 , k ( Y t − k · 48 ) + f 1 ( DayType t , Offset t ) + � 12 i = 1 f 2 , i ( T t − 12 ) 1 { M t = i } + f 3 ( ToY t ) + f 4 ( t 15 ) + f 5 ( t )+ f 6 ( Cloud t ) + f 7 ( T t ) + f 8 ( W t ) + f 9 ( θ t ) + f 10 ( θ Min ) + f 11 ( θ Max ) + ǫ t , t t where at time t : Y t is the electric demand ToY t is the time of the year of observation t DayType t and Offset t are categorical variables indicating the type of day and the daylight saving time M t is the Month, t 15 = t 1 { T t ≤ 15 } estimating a heating trend several lagged and smoothed variables related to temperature T t and θ t an exponential smoothing of T t , Cloud t and W t are the cloud cover and the wind Jean Michel Poggi Nonstationary time series forecasting and functional clustering using wavelets
Industrial motivation Wavelets to describe Functional Data Electricity demand data The Kernel+Wavelet+Functional (KWF) model Electricity demand forecasting How to cope with non stationarities Aim Clustering functional data Functional time series Clustering individual data Methods to design the forecasting model Short-term electricity demand forecast in literature A lot of methods are available to build prediction models [Weron (2007)] The most classical models, including those of the SARIMAX family, constitute an important baseline and are a favourite time series model They can achieve excellent results even if the price to pay is sometimes the complexity (a lot of parameters to estimate) and a certain difficulty to be adaptive. An interesting and conceptually simple extension of these models is to consider additive non linear models. Regression: Of course, today black box-type models forgetting the interpretation of the role of variables in favor of the sole objective of forecasting are particularly in vogue, thanks to the machine learning era Several types of methods coexist, but one that is most frequently used is undoubtedly neural networks [Park et al. (2011)] , mainly used by engineers and computer scientists, with its last (complex) avatar: deep learning Jean Michel Poggi Nonstationary time series forecasting and functional clustering using wavelets
Industrial motivation Wavelets to describe Functional Data Electricity demand data The Kernel+Wavelet+Functional (KWF) model Electricity demand forecasting How to cope with non stationarities Aim Clustering functional data Functional time series Clustering individual data Electricity demand forecast Short-term electricity demand forecast in literature Time series analysis: sarima(x) , Kalman filter [Dordonnat et al. (2009)] Machine learning [Devaine et al. (2010)] Similarity search based methods [Poggi (1994), Antoniadis et al. (2006)] Regression: edf modelisation scheme [Bruhns et al. (2005)] , gam [Pierrot and Goude (2011)] , Bayesian approach [Launay, Philippe and Lamarche (2012)] New challenges Market liberalization: may produce variations on clients’ perimeter that risk to induce nonstationarities on the signal. Development of smart grids and smart meters. But, almost all the models rely on a monoscale representation of the data. Jean Michel Poggi Nonstationary time series forecasting and functional clustering using wavelets
Industrial motivation Wavelets to describe Functional Data Electricity demand data The Kernel+Wavelet+Functional (KWF) model Electricity demand forecasting How to cope with non stationarities Aim Clustering functional data Functional time series Clustering individual data FD as slices of a continuous process [Bosq, (1990)] The prediction problem Suppose one observes a square integrable continuous-time stochastic process X = ( X ( t ) , t ∈ R ) over the interval [ 0 , T ] , T > 0; We want to predict X all over the segment [ T , T + δ ] , δ > 0 Divide the interval into n subintervals of equal size δ . Consider the functional-valued discrete time stochastic process Z = ( Z k , k ∈ N ) , where N = { 1 , 2 , . . . } , defined by X t t 0 T T + δ Jean Michel Poggi Nonstationary time series forecasting and functional clustering using wavelets
Industrial motivation Wavelets to describe Functional Data Electricity demand data The Kernel+Wavelet+Functional (KWF) model Electricity demand forecasting How to cope with non stationarities Aim Clustering functional data Functional time series Clustering individual data FD as slices of a continuous process [Bosq, (1990)] The prediction problem Suppose one observes a square integrable continuous-time stochastic process X = ( X ( t ) , t ∈ R ) over the interval [ 0 , T ] , T > 0; We want to predict X all over the segment [ T , T + δ ] , δ > 0 Divide the interval into n subintervals of equal size δ . Consider the functional-valued discrete time stochastic process Z = ( Z k , k ∈ N ) , where N = { 1 , 2 , . . . } , defined by X t Z 3 ( t ) Z 4 ( t ) Z 6 ( t ) Z k ( t ) = X ( t + ( k − 1 ) δ ) Z 1 ( t ) Z 2 ( t ) Z 5 ( t ) k ∈ N ∀ t ∈ [ 0 , δ ) t 0 1 δ 2 δ 3 δ 4 δ 5 δ 6 δ T + δ If X contents a δ − seasonal component, Z is particularly fruitful. Jean Michel Poggi Nonstationary time series forecasting and functional clustering using wavelets
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