Nonstationary time series forecasting and functional clustering - - PowerPoint PPT Presentation

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

1

Industrial motivation

2

The Kernel+Wavelet+Functional (KWF) model

3

How to cope with non stationarities

4

Clustering functional data Scale-oriented feature extraction Wavelet coherence based dissimilarity

5

Clustering individual data

Jean Michel Poggi Nonstationary time series forecasting and functional clustering using wavelets

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 Electricity demand data Electricity demand forecasting Aim Functional time series

Outline

1

Industrial motivation

2

The Kernel+Wavelet+Functional (KWF) model

3

How to cope with non stationarities

4

Clustering functional data Scale-oriented feature extraction Wavelet coherence based dissimilarity

5

Clustering individual data

Jean Michel Poggi Nonstationary time series forecasting and functional clustering using wavelets

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 Electricity demand data Electricity demand forecasting Aim Functional time series

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 The Kernel+Wavelet+Functional (KWF) model How to cope with non stationarities Clustering functional data Clustering individual data Electricity demand data Electricity demand forecasting Aim Functional time series

Electricity demand forecasting (general)

Short-term electricity demand: a classical model Yt+1 = F(Yt, Yt−k, . . . , Yt−K; Xt, Xt−k′, . . . , Xt−K′; Ct) + 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

• ne 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 The Kernel+Wavelet+Functional (KWF) model How to cope with non stationarities Clustering functional data Clustering individual data Electricity demand data Electricity demand forecasting Aim Functional time series

Electricity demand forecasting (general)

Short-term electricity demand: an additive model Yt = K

k=1 f0,k(Yt−k·48) + f1(DayTypet, Offsett) +

12

i=1 f2,i(Tt−12)1{Mt=i} + f3(ToYt) + f4(t15) + f5(t)+

f6(Cloudt) + f7(Tt) + f8(Wt) + f9(θt) + f10(θMin

t

) + f11(θMax

t

) + ǫt, where at time t: Yt is the electric demand ToYt is the time of the year of observation t DayTypet and Offsett are categorical variables indicating the type of day and the daylight saving time Mt is the Month, t15 = t1{Tt≤15} estimating a heating trend several lagged and smoothed variables related to temperature Tt and θt an exponential smoothing of Tt, Cloudt and Wt 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 The Kernel+Wavelet+Functional (KWF) model How to cope with non stationarities Clustering functional data Clustering individual data Electricity demand data Electricity demand forecasting Aim Functional time series

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 The Kernel+Wavelet+Functional (KWF) model How to cope with non stationarities Clustering functional data Clustering individual data Electricity demand data Electricity demand forecasting Aim Functional time series

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 The Kernel+Wavelet+Functional (KWF) model How to cope with non stationarities Clustering functional data Clustering individual data Electricity demand data Electricity demand forecasting Aim Functional time series

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 = (Zk, k ∈ N), where N = {1, 2, . . .}, defined by Xt t T T + δ

Jean Michel Poggi Nonstationary time series forecasting and functional clustering using wavelets

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 Electricity demand data Electricity demand forecasting Aim Functional time series

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 = (Zk, k ∈ N), where N = {1, 2, . . .}, defined by Xt t 1δ 2δ 3δ 4δ 5δ 6δ T + δ

Z1(t) Z2(t) Z5(t) Z3(t) Z4(t) Z6(t)

Zk(t) = X(t + (k − 1)δ) k ∈ N ∀t ∈ [0, δ) If X contents a δ−seasonal component, Z is particularly fruitful.

Jean Michel Poggi Nonstationary time series forecasting and functional clustering using wavelets

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 Electricity demand data Electricity demand forecasting Aim Functional time series

Prediction of functional time series Let (Zk, k ∈ Z) be a stationary sequence of H-valued r.v. Given Z1, . . . , Zn we want to predict the future value of Zn+1. A predictor of Zn+1 using Z1, Z2, . . . , Zn is

• Zn+1 = E[Zn+1|Zn, Zn−1, . . . , Z1].

Autoregressive Hilbertian process of order 1 The arh(1) centred process states that at each k, Zk = ρ(Zk−1) + ǫk (1) where ρ is a compact linear operator and {ǫk} an H−valued strong white noise. Under mild conditions, equation (1) has a unique solution which is a strictly stationary process with innovation {ǫk}k∈Z. [Bosq, (1991)] When Z is a zero-mean arh(1) process, the best predictor of Zn+1 given {Z1, . . . , Zn−1} is:

• Zn+1 = ρ(Zn).

Jean Michel Poggi Nonstationary time series forecasting and functional clustering using wavelets

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 Electricity demand data Electricity demand forecasting Aim Functional time series

Overview

key an appropriate distance between current and past situations. idea 1 Similar past causes produce similar future consequences. idea 2 Similar shapes form one class.

Jean Michel Poggi Nonstationary time series forecasting and functional clustering using wavelets

Wavelets to cope with fd

domain-transform technique for hierarchical decomposing finite energy signals description in terms of a broad trend (approximation part), plus a set of localized changes kept in the details parts.

Discrete Wavelet Transform If z ∈ L2([0, 1]) we can write it as z(t) =

2j0 −1

• k=0

cj0,kφj0,k(t) +

∞

• j=j0

2j −1

• k=0

dj,kψj,k(t), where cj,k =< g, φj,k >, dj,k =< g, ϕj,k > are the scale coefficients and wavelet coefficients respectively, and the functions φ et ϕ are associated to a

• rthogonal mra of L2([0, 1]).

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 Prediction algorithm

Outline

1

Industrial motivation

2

The Kernel+Wavelet+Functional (KWF) model

3

How to cope with non stationarities

4

Clustering functional data Scale-oriented feature extraction Wavelet coherence based dissimilarity

5

Clustering individual data

Jean Michel Poggi Nonstationary time series forecasting and functional clustering using wavelets

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 Prediction algorithm

Approximation and details

In practice, we don’t dispose of the whole trajectory but only with a (possibly noisy) sampling at 2J points, for some integer J. Each approximated segment Zi,J(t) is broken up into two terms: a smooth approximation Si(t) (lower freqs) a set of details Di(t) (higher freqs) Zi,J(t) =

2j0 −1

• k=0

c(i)

j0,kφj0,k(t)

• Si (t)

+

J−1

• j=j0

2j −1

• k=0

d(i)

j,kψj,k(t)

• Di (t)

The parameter j0 controls the separation. We set j0 = 0.

• zJ(t) = c0φ0,0(t) +

J−1

• j=0

2j −1

• k=0

dj,kψj,k(t).

Jean Michel Poggi Nonstationary time series forecasting and functional clustering using wavelets

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 Prediction algorithm

A two step prediction algorithm

Step I: Dissimilarity between segments Search the past for segments that are similar to the last one. For two observed series of length 2J say Zm and Zl we set for each scale j ≥ j0: distj(Zm, Zl) =

2j −1

• k=0

(d(m)

j,k − d(l) j,k)2

1/2

Then, we aggregate over the scales taking into account the number of coefficients at each scale D(Zm, Zl) =

J−1

• j=j0

2−j/2distj(Zm, Zl)

Jean Michel Poggi Nonstationary time series forecasting and functional clustering using wavelets

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 Prediction algorithm

A two step prediction algorithm

Step 2: Kernel regression Obtain the prediction of the wavelet coefficients Ξn+1 = {c(n+1)

J,k

, d(n+1)

j,k

: k = 0, 1, . . . , 2j − 1} for Zn+1

• Ξn+1 =

n−1

• m=1

wm,nΞm+1 wm,n = K D(Zn,Zm)

hn

• n−1

m=1 K D(Zn,Zm) hn

• Finally, the prediction of Zn+1 is obtained through the inverse DWT.

Jean Michel Poggi Nonstationary time series forecasting and functional clustering using wavelets

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 Prediction algorithm

Daily prediction error

Figure : Daily prediction error (in mapex100). Huge problem during the cold season Large prediction errors at quite regular frequency during warm season Need of corrections to deal with non stationarities.

Jean Michel Poggi Nonstationary time series forecasting and functional clustering using wavelets

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 Prediction algorithm

Let us predict Saturday 10 September 2005

We use Antoniadis et al., (2006) prediction method with corrections to cope with non stationarity. Use the last observed segment (n = 9/Sept/2005) to look for similar segments in past. Construct a similarity index SimilIndex (using a kernel). Prediction can be written as

• Loadn+1(t) =

n−1

• m=1

SimilIndexm,n × Loadm+1(t) First difference correction of the approximation part. Use of groups to anticipate calendar transitions.

Jean Michel Poggi Nonstationary time series forecasting and functional clustering using wavelets

SimilIndex date SimilIndex 2004-09-10 0.455 2003-09-05 0.141 2002-09-06 0.083 2004-09-03 0.070 2003-09-19 0.068 2000-09-08 0.058 2000-09-15 0.019 1999-09-10 0.017 similar past similar future

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 Corrections to handle nonstationarity

Outline

1

Industrial motivation

2

The Kernel+Wavelet+Functional (KWF) model

3

How to cope with non stationarities

4

Clustering functional data Scale-oriented feature extraction Wavelet coherence based dissimilarity

5

Clustering individual data

Jean Michel Poggi Nonstationary time series forecasting and functional clustering using wavelets

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 Corrections to handle nonstationarity

Corrections to handle nonstationarity

On mean level base

• Sn+1(t) = n−1

m=1 wm,nSm+1(t)

diff

• Sn+1(t) = Sn(t) + n−1

m=2 wm,n∆(Sm)(t)

Figure : Daily prediction error (in mapex100).

Jean Michel Poggi Nonstationary time series forecasting and functional clustering using wavelets

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 Corrections to handle nonstationarity

Corrections to handle nonstationarity

On groups by post-treatment Define new weights and renormalize. ˜ wm,n =

• ww,m

if gr(m) = gr(n)

• therwise

gr(n) is the group of the n-th segment.

1

Deterministic groups: Calendar or Calendar transitions.

2

Groups coming from clustering analysis. (e.g. temperature curves)

3

Cross deterministic with clustering groups (e.g. calendar-temperature

transitions).

Figure : Daily prediction error (in mapex100).

Jean Michel Poggi Nonstationary time series forecasting and functional clustering using wavelets

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 Scale-oriented feature extraction Wavelet coherence based dissimilarity

Outline

1

Industrial motivation

2

The Kernel+Wavelet+Functional (KWF) model

3

How to cope with non stationarities

4

Clustering functional data Scale-oriented feature extraction Wavelet coherence based dissimilarity

5

Clustering individual data

Jean Michel Poggi Nonstationary time series forecasting and functional clustering using wavelets

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 Scale-oriented feature extraction Wavelet coherence based dissimilarity

Clustering functional data

Segmentation of X may not suffice to satisfy the stationarity hypothesis. If a grouping effect exists, we may considered stationary within each group. Conditionally on the grouping, functional time series prediction methods can be applied. We propose a clustering procedure that discover the groups from a bunch of curves. We use wavelet transforms to take into account the fact that curves may present non stationary patterns. Two strategies to cluster functional time series:

1

Feature extraction (summary measures of the curves).

2

Direct similarity between curves.

Jean Michel Poggi Nonstationary time series forecasting and functional clustering using wavelets

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 Scale-oriented feature extraction Wavelet coherence based dissimilarity

Energy decomposition of the DWT

Energy conservation of the signal z2 ≈ zJ2

2 = c2 0,0 + J−1

• j=0

2j −1

• k=0

d2

j,k = c2 0,0 + J−1

• j=0

dj2

2.

For each j = 0, 1, . . . , J − 1, we compute the absolute and relative contribution representations by contj = ||dj||2

• AC

and relj = ||dj||2

• j ||dj||2
• RC

. They quantify the relative importance of the scales to the global dynamic. RC normalizes the energy of each signal to 1.

Jean Michel Poggi Nonstationary time series forecasting and functional clustering using wavelets

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 Scale-oriented feature extraction Wavelet coherence based dissimilarity

Schema of procedure

• 0. Data preprocessing. Approximate sample paths of z1(t), . . . , zn(t)
• 1. Feature extraction. Compute either of the energetic components using absolute

contribution (AC) or relative contribution (RC).

• 2. Feature selection. Screen irrelevant variables.

[Steinley & Brusco (’06)]

• 3. Determine the number of clusters. Detecting significant jumps in the transformed

distortion curve.

[Sugar & James (’03)]

• 4. Clustering. Obtain the K clusters using PAM algorithm.

Jean Michel Poggi Nonstationary time series forecasting and functional clustering using wavelets

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 Scale-oriented feature extraction Wavelet coherence based dissimilarity

Toy example

Figure : On the left, some typical simulated trajectories of the sinus model (top panel), the far1 model (middle), and the far2 model (bottom). On the right, the mean scales’ energy absolute contribution by cluster.

Jean Michel Poggi Nonstationary time series forecasting and functional clustering using wavelets

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 Scale-oriented feature extraction Wavelet coherence based dissimilarity

Clustering Feature Extraction Raw curves Abbreviation RC RAW Mean Global error 21.05 (4.125) 25.32 (4.834) Mean Rand Index 0.414 (0.092) 0.335 (0.109) Table : Indicators of the clustering quality. Mean values over the 100 replicates with standard deviation between parenthesis. Figure : Boxplots of the misclassification error (left) and the Adjusted Rand Index for the 100 replicates of the simulated data set. Clustering using PAM on the extracted features (left) and on the raw curves (right).

Jean Michel Poggi Nonstationary time series forecasting and functional clustering using wavelets

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 Scale-oriented feature extraction Wavelet coherence based dissimilarity

EDF data

Figure :

French electricity power demand on autumn (top left), winter (bottom left), spring (top right) and summer (bottom right). Feature extraction: The significant scales are independent of the number of clusters. Significant scales are associated to mid-frequencies. The retained scales parametrize cycles of 1.5, 3 and 6 hours (AC).

Jean Michel Poggi Nonstationary time series forecasting and functional clustering using wavelets

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 Scale-oriented feature extraction Wavelet coherence based dissimilarity

Figure :

Number of clusters by feature extraction of the AC (top). From left to right: distortion curve, transformed distortion curve and first difference on the transformed distortion curve.

Jean Michel Poggi Nonstationary time series forecasting and functional clustering using wavelets

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 Scale-oriented feature extraction Wavelet coherence based dissimilarity

(a) Curves (b) Calendar Figure : Curves membership of the clustering using ac based dissimilarity (a) and the corresponding calendar positioning (b).

Jean Michel Poggi Nonstationary time series forecasting and functional clustering using wavelets

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 Scale-oriented feature extraction Wavelet coherence based dissimilarity

An alternative: function-based distance

Distance based on wavelet-correlation between two time series Can be used to measure relationship between two functions The strength of the relation is hierarchically decomposed across scales without losing of time location Drawback: needs more computation time and storage (complex values)

Jean Michel Poggi Nonstationary time series forecasting and functional clustering using wavelets

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 Scale-oriented feature extraction Wavelet coherence based dissimilarity

CWT

Continuous WT Starting with a mother wavelet ψ consider ψa,τ = a−1/2ψ t−τ

a

• .

The CWT of a function z ∈ L2(R) is, Wz(a, τ) =

∞

−∞

z(t)ψ∗

a,τ(t)dt

As for Fourier transform, a spectral approach is possible. Sz(a, τ) = |Wz(a, τ)|2 wavelet spectrum Wz,x(a, τ) = Wz(a, τ)W ∗

x (a, τ)

cross-wavelet transform

Jean Michel Poggi Nonstationary time series forecasting and functional clustering using wavelets

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 Scale-oriented feature extraction Wavelet coherence based dissimilarity

Wavelet coherence

R2

z,x(a, τ) =

| ˜ Wx,y(a, τ)|2 | ˜ Wx,x(a, τ)|| ˜ Wy,y(a, τ)| , Based on the extended R2 coefficient, we can construct an coefficient of determination between two wavelet spectrums WER2

z,x =

∞ ∞

−∞ | ˜

Wz,x(a, τ)|dτ

2

da

∞ ∞

−∞ | ˜

Wz,z(a, τ)|dτ ∞

−∞ | ˜

Wx,x(a, τ)|dτ

• da

. And obtain a dissimilarity based on it d(z, x) =

• JN(1 −

WER

2 z,x)

Jean Michel Poggi Nonstationary time series forecasting and functional clustering using wavelets

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 Scale-oriented feature extraction Wavelet coherence based dissimilarity

Wavelet coherence

We proceed as follows: Transform data z1(t), . . . , zn(t) using the CWT and Morlet wavelet to

• btain n matrices of size J × N.

Compute a dissimilarity matrix with the coherency based dissimilarity. Using PAM obtain clusters k = 8 clusters. Rand Index (AC, WER) = 0.26

Jean Michel Poggi Nonstationary time series forecasting and functional clustering using wavelets

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 Scale-oriented feature extraction Wavelet coherence based dissimilarity Jean Michel Poggi Nonstationary time series forecasting and functional clustering using wavelets

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

1

Industrial motivation

2

The Kernel+Wavelet+Functional (KWF) model

3

How to cope with non stationarities

4

Clustering functional data Scale-oriented feature extraction Wavelet coherence based dissimilarity

5

Clustering individual data

Jean Michel Poggi Nonstationary time series forecasting and functional clustering using wavelets

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

Disaggregated electricity data

Data set of 25011 professional customers Sampling rate: 30 minutes Period: 2009, 2010 and 2011 (only 6 month) 1 year ∼ 438 millions records ∼ 3.25 Go Figure : Aggregate demand (left) and individual demand (right) for 2010 (with infraday filtering) .

Jean Michel Poggi Nonstationary time series forecasting and functional clustering using wavelets

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

Clients hierarchical structure and prediction

Figure : Hierarchical structure of N individual clients among K groups Zt Zt+1

   

Zt,1 Zt,2 . . . Zt,K

       

Zt+1,1 Zt+1,2 . . . Zt+1,k

   

Zt: aggregate demand at t Zt,k: demand of group k at time t Groups can express tariffs, geographical dispersion, client class ... Profiling vs Prediction We follow Misiti et al. (2010) to construct clusters of customers to better predict the aggregate.

Jean Michel Poggi Nonstationary time series forecasting and functional clustering using wavelets

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

2-steps strategy for building sequence of partitions

1st step: create a large number of super customers (K ′ = 200) 2nd step: after aggregation, compute the aggregates

Jean Michel Poggi Nonstationary time series forecasting and functional clustering using wavelets

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

Prediction performance along sequences of partitions

1- Build sequences of consumer classes on 2009 data 2- Select by measuring the quality of daily KWF forecasts throughout 2010 Figure : Mean absolute prediction error (MAPE) as a function of the number

• f clusters for the baseline (without clustering) and two clustering variants

Jean Michel Poggi Nonstationary time series forecasting and functional clustering using wavelets

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

Aggregate signals are KWF-predictable

Figure : Aggregate signals for 2 clusters Figure : Aggregate signals for 3 clusters

Jean Michel Poggi Nonstationary time series forecasting and functional clustering using wavelets

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

References

[1] A. Antoniadis, X. Brossat, J. Cugliari, and J.-M. Poggi (2013) Functional Clustering using Wavelets. International Journal of Wavelets, Multiresolution and Information Processing, 11(1). [2] A. Antoniadis, X. Brossat, J. Cugliari, and J.-M. Poggi (2012) Prévision d’un processus à valeurs fonctionnelles en présence de non stationnarités. Application à la consommation d’électricité. Journal de la Société Française de Statistique, 153(2) :52 – 78. [3] A. Antoniadis, X. Brossat, J. Cugliari, and J.-M. Poggi (2014) Une approche fonctionnelle pour la prévision non-paramétrique de la consommation d’électricité. Journal de la Société Française de Statistique, 155(2) :202 – 219. [4] M. Misiti, Y. Misiti, G. Oppenheim, and J. M. Poggi (2010) Optimized Clusters for Disaggregated Electricity Load Forecasting. REVSTAT – Statistical Journal, 8(2):105 – 124. [5] J. Cugliari, Y. Goude and J.-M. Poggi (2016) Disaggregated Electricity Forecasting using Wavelet-Based Clustering of Individual Consumers. Proc. IEEE EnergyCon 2016, KU Leuven, 4-8 April 2016, 6 pages.

Jean Michel Poggi Nonstationary time series forecasting and functional clustering using wavelets