Time Series Forecasting using CCA and Kohonen Maps - Application to Electricity Consumption

Transcription

1 ESANN'2000 proceedings - European Symposium on Arificial Neural Neworks Bruges (Belgium), April 2000, D-Faco public., ISBN , pp Time Series Forecasing using CCA and Kohonen Maps - Applicaion o Elecriciy Consumpion A. Lendasse 1, J. Lee 2, V. Werz 1, M. Verleysen 2 1 Universié caholique de Louvain, CESAME, 4 av. G. Lemaîre B-1348 Louvain-la-Neuve, Belgium, lendasse@auo.ucl.ac.be. 2 Universié caholique de Louvain, Elecriciy Dep., 3 pl. du Levan, B-1348 Louvain-la-Neuve, Belgium, {lee, verleysen}@dice.ucl.ac.be. Absrac. A general-purpose useful parameer in ime series forecasing is he regressor, corresponding o he minimum number of variables necessary o forecas he fuure values of he ime series. If he models used are non linear, he choice of his regressor becomes very difficul. We will show a quasiauomaic mehod using Curvilinear Componen Analysis o build i. This mehod will be applied o elecric consumpion of Poland. 1. Inroducion Time series forecasing is a grea challenge in many fields of applicaion. The financial ones wan o forecas sock exchange courses or indices of sock markes; daa processing specialiss: flow of informaion on heir neworks; producers of elecriciy: he consumpion of he following day. The common poin o heir problems is he following: how o analyse and use he pas o forecas he fuure? Many echniques exis, such as for example he linear mehods (ARX, ARMA...) [1,2] and also he non linear mehods as arificial neural neworks [3]. In general, hese mehods ry o build a model of he process, which one wans o forecas. This model connecs he las values of he series o hese fuure values. The common difficuly o all hese mehods is he deerminaion of sufficien and necessary informaion o a good forecasing. If informaion is insufficien, he forecasing will be poor. If on he conrary, informaion is useless or redundan, modelling will be difficul or even skewed. In his paper, we will describe an original mehod for he deerminaion of informaion useful o a good forecasing. We also will briefly presen a lile known model of non linear forecasing using Kohonen s Maps. And finally, we will illusrae he presened mehods by a radiional example of ime series, he forecasing of he elecric consumpion of a counry. The daa, which we will use, come from Poland.

2 ESANN'2000 proceedings - European Symposium on Arificial Neural Neworks Bruges (Belgium), April 2000, D-Faco public., ISBN , pp Non linear Mehods for Forecasing We will briefly describe he general mehod of non linear forecasing [4]. Tha is o say a series, which is lengh N. We noe i y wih variable beween 1 and N. The model, which is usually used o collec he dynamics of he process, is he following: y + 1 = f (y, y 1,..., y n, ϑ), (1) where ϑ represens he whole of he parameers which make i possible he model F o approximae as well as possible he saring series. The vecor y o y -n is called regressor. I is obvious ha he choice of he regressor and hus N is capial. If his one is badly done, he model will be vague or possibly skewed. In he bes case, he model will be righ bu he deerminaion of ϑ will be very difficul. Several mehods exis o choose he regressor. For examples, o use he opimal regressor of a linear model or o use mehods called pruning. Bu hese mehods are oo simple or require oo many calculaions. 3. Deerminaion of he Regressor using CCA The mehod ha we will presen is raher differen. Indeed, we will no selec he bes regressor bu build i by a non linear projecion. Here how we will proceed. Firsly, le us build a regressor of grea dimension, which will conain oo much informaion: Y [ y, y,..., y ] =, (2) 1 We hus creaed a space wih N dimensions in which informaion is redundan. This redundancy of informaion can be expressed in he following way: he real or inrinsic dimension of space is lower han N. Le us call d his real dimension. Anoher manner of expressing his is: he y daa form a d-dimensional surface in space R n. We hus are going o ried o build a new regressor who is of dimension d and which sores all he informaion conained in he iniial regressor. For his purpose, a projecion can be used. Various echniques of projecion exis o pass from a space of dimension n o a space of dimension d. For example, Principal Componen Analysis (PCA), bu his one is perhaps no judicious because i is a linear projecion. An ineresing alernaive of he PCA is he Curvilinear Componen Analysis (CCA), which is one of is non linear exensions [5,6,7]. Here an example of projecion carried ou by CCA: n Fig. 1: Projecion carried ou by CCA from R 3 o R 2.

3 ESANN'2000 proceedings - European Symposium on Arificial Neural Neworks Bruges (Belgium), April 2000, D-Faco public., ISBN , pp We can hus summarise he oal mehod: he regressor: Y [ y, y,..., y ] =, (3) 1 n is projeced using CCA o: [ z,z,..., z ] Z =, (4) 1 2 d And finally, he model of forecasing is buil: 4. Arificial Example y + 1 = f (z1,z2,...,zz, ϑ), (5) The following figure shows us an arificial series on which we will es he presened mehod. Fig. 2: An arificial ime series. The saring regressor ha we will choose is of dimension 3. The following figure represens he regressor a every momen of he series. Fig. 3: The iniial regressor a every momen of he series.

4 ESANN'2000 proceedings - European Symposium on Arificial Neural Neworks Bruges (Belgium), April 2000, D-Faco public., ISBN , pp I is clearly visible on he figure ha he inrinsic dimension of he series is 1. We hus will projec he regressor o R using CCA. Wih he following figure, we represen y +1 according o new regressor Z. Fig. 4: y +1 according o new regressor Z. This series can be easily modelled using a RBF nework (Radial Basis Funcion Nework) wih 5 gaussian kernels [8]. The MSE (Mean Square Error) obained is In comparison he MSE obained wih he basic regressor is: 0.26 wih a linear model and 0.11 using a RBF wih 25 gaussian kernels. 5. Forecasing Time Series using Kohonen Maps The model of forecasing ha we will describe is a lile known model and is based on he self-organizing maps of Kohonen (SOM) [9]. Le us suppose ha we have a good regressor: Z. We will connec his regressor wih his following value y +1 : [ z,z,...,z, y ] x 1 2 d + 1 =, (6) Then, we will quanify he space formed wih he x by a SOM whose cenroids will be noed C i. These cenroids are hus made of wo pars, he firs which comes from he regressor: C i1 and in second par he forecasing: C i2. These cenroids form our model. Indeed, a a momen our forecasing will be calculaed in he following way: firsly calculaion of he regressor z, hen we seek he C i1 par of cenroid ha is closes o z. The forecasing is he corresponding C i2 of his closes cenroid. 6. Applicaion o Elecriciy Consumpion The series ha we will sudy represens he average day labourer of elecric consumpion in Poland [10]. As one can see i on fig. 5, his series was sandardized.

5 ESANN'2000 proceedings - European Symposium on Arificial Neural Neworks Bruges (Belgium), April 2000, D-Faco public., ISBN , pp One sees well here he seasonal variaion which is quasi sinusoidal. If we looks on a few week scale (fig. 6), i is seen ha he elecric consummaion has he shape of eeh of saw, maximum he days of he week and minimal he weekend. Fig. 5: Elecric consumpion in Poland. Fig. 6: Elecric consumpion in Poland on a few week scales. The series is divided in a raining se (wo hird of he daa) and a es se (one hird of he daa). If we use a linear model, a very good regressor is ha which is rained of he eigh preceding values of he series. This iniial regressor is hen projeced o a 4- dimensional space using CCA. We hen quanify he space formed wih he projecion

6 ESANN'2000 proceedings - European Symposium on Arificial Neural Neworks Bruges (Belgium), April 2000, D-Faco public., ISBN , pp by a SOM wih 20x20 cenroids. The MSE obained wih he es se is If we had kep he saring regressor he MSE would be Because of he inaccuracies during projecion, he improvemen is less han ha we waied bu i remains significan. We mus poin ou ha calculaion ime is roughly divided by wo. 7. Conclusion The preliminary resuls presened in his paper show ha he CCA mehod can be used o esimae he inrinsic dimension of a daa se, and hen he bes regressor in he problem of imes series forecasing. This mehod avoids over raining. Progress is sill o realize in CCA; indeed inaccuracies in his one limi he improvemen of he performances in he forecasing. References [1] Box G.E.P., Jenkins G.: Time Series analysis: Forecasing and Conrol Cambridge Universiy Press, [2] Ljung L.: Sysem Idenificaion - Theory for User, Prenice-Hall, [3] Weigend A. S., Gershenfeld N.A.: Times Series Predicion: Forecasing he fuure and Undersanding he Pas, Addison-Wesley Publishing Company, [4] Sjoberg J., Zhang Q., Ljung L. e al., Non linear black box modelling in sysem idenificaion: A unified overview, Auomaica 33: (6) , [5] Demarines P., Heraul J., Curvilinear componen analysis: A selforganizing neural nework for non linear mapping of daa ses IEEE Transacion on Neural Neworks 8: (1) January [6] Demarines P. and Héraul J. Vecor Quanizaion and Projecion In A. Prieo, J. Mira, J. Cabesany, edior, Inernaional Workshop on Arificial Neural Neworks, Vol. 686 of Lecure Noes in Compuer Sciences, pp Springer-Verlag, [7] Demarine P., Analyse de données par réseaux de neurones auoorganisés. Thèse présenée en vue de l obenion du grade de Doceur de l Insiu Naional Polyechnique de Grenoble [8] 9HUOH\VHQ 0 +ODYDþNRYD. ³An Opimised RBF Nework for Approximaion of Funcions. In: Proc of European Symposium on Arificial Neural Neworks, Brussels (Belgium), April [9] Kohonen T., Self-organising Maps, Springer Series in Informaion Sciences, Vol. 30, Springer, Berlin, [10] Corell M, Girard B, Girard Y, e al. Daily elecrical power curves: Classificaion and forecasing using a Kohonen map, Naural o Arificial Neural Compuaion 930: , 1995.