CONSTRUCTIVE NEURAL NETWORKS IN FORECASTING WEEKLY RIVER FLOW

Transcription

1 CONSTRUCTIVE NEURAL NETWORKS IN FORECASTING WEEKLY RIVER FLOW MÊUSER VALENÇA Universidade Salgado de Oliveira - UNIVERSO /Comanhia Hidroelétrica do São Francisco - Chesf Rua Gasar Peres, 47/4 CDU, Recife, Pernambuco, Brazil meuser@elogicacombr TERESA LUDERMIR Universidade Federal de Pernambuco Deartamento de Informática Cx Postal 785, Recife, Brazil tbl@cinufebr Abstract This aer resents an constructive neural network model for seasonal streamflow forecasting This Surface water hydrology is basic to the design and oeration of the reservoir A good examle is the oeration of a reservoir with an uncontrolled inflow but having a means of regulating the outflow If information on the nature of the inflow is determinable in advance, then the reservoir can be oerated by some decision rule to minimize downstream flood damage For this reasons, several comanies in the Brazilian Electrical Sector use the linear time-series models such as PARMA (Periodic Auto regressive Moving Average) models develoed by Box-Jenkins This aer rovides for river flow rediction a numerical comarison between neural networks, called non-linear sigmoidal regression Blocks networks (NSRBN) and PARMA models The model was imlemented to forecast weekly average inflow on an ste-ahead basis It was tested on four hydroelectric lants located in different river basins in Brazil The results obtained in the evaluation of the erformance of NSRBN were better than the results obtained with PARMA models Introduction Models for river flow forecast are a fundamental tool in water resource studies, since they are in change of establishing future reservoir water inflow These redictions are of central imortance in the lanning of a water resource system, being resonsible for the otimization of the system as a whole Among the tradition techniques used for this urose, we highlight concetual models for simulation and the linear time-series models, such as ARIMA (Auto regressive Integrated Moving Average) models, develoed by Box-Jenkins, 976 [] Concetual models are designed to reresent the general internal sub-rocesses and hysical mechanisms which govern the hydrologic cycle While these models ignore simle asects such as satially distributed and time-varying, they attemt to incororate realistic reresentations of the major noninearities inherent in the rainfall- runoff relationshis[] However, the imlementation and calibration of such a model can tyically resent various difficulties, requiring sohisticated mathematical tool, significant amounts of calibration data and some degree of exertise and exerience with resect to the model For this reasons, several comanies in the Brazilian Electrical Sector use the linear time-series models such as ARMA(AutoRegressive Moving Average)models develoed by Box-Jenkins These models are relatively easy to develo and imlement and they have been found to rovide satisfactory redictions in many alications In this aer, we resent a new class of higher-order feedforward neural networks, called non-linear sigmoidal regression blocks networks (NSRBN)[3] and demonstrates the otential of such models for streamflow forecasting The NSRBN model aroach is show to rovide better results to the forecast of weekly streamflow the five hydroelectric ower lant, art of the Brazilian Electrical Sector, than the linear PARMA models aroach Section brings an overview of the NSRBN algorithms, followed by a brief resentation of PARMA model in the section 3 Section 4 shows an evaluation of our results Finally, section 5 concludes the aer Constructive Neural Networks Proceedings of the 4th International Conference on Comutational Intelligence and Multimedia Alications (ICCIMA')

2 Network Architecture Neural network architectures are either fixed or dynamic in size In a fixed-size architecture, the size of an instance of a network, in terms of number of units and weights, does not change from its first construction On the other hand, in a dynamic-size architecture, the size of a network can be changed during the learning rocess In articular, a constructive learning method grows the network structure from a small, basic network for a given roblem and the network becomes larger as learning roceeds until the desired level of aroximation error is attained The main advantage of a constructive learning algorithm is that the networks realize a arsimonious aroximation of the roblem, tyically resulting in better generalization[4] Figure shows non-linear sigmoidal regression blocks networks (NSRBN) with a single outut This network is a fully connected two-layered feedforward network Let x = [,x,x,x 3,,x N ] T be an N+- dimensional augmented inut column vector where x i denotes the ith comonent of x The inuts are weighted by N+-dimensional weight vectors w = [w h,w h,w h,,w Nh ] T, h=,,, where is the desired degree[5] x st degree f x nd degree f f x 3 th degree f f d x N dth degree Figure - NSRBN (network architecture) Fig shows a generic block architecture The outut of the block, f, is given by = a ( σ + a ( σ net( ) net() + θ ) + θ ) + a ( σ net() + θ ) + () where σ net(h) is a suitable activation function, net(h) = w x ( net( h) = w x + w, h=,,, ), θ h is an adjustable threshold and a = [a,a,a 3,, a ] T is an weight vectors from hidden layer T h N i = ih i h x x W i,h a a f x N a Figure - Block architecture Proceedings of the 4th International Conference on Comutational Intelligence and Multimedia Alications (ICCIMA')

3 Note that the block has an nonlinear function in the outut layer Thus, an unknown scalar function f in R d can be aroximated by the direct use of the NSRBN of degree u to d based on d net( o) = = σ ( ) () where σ net(o) is a suitable activation function of outut Learning Algorithm for the NSRBN The constructive algorithm for NSRBN is based on the constructive learning method The rincile can be formulated as follows: when the model comlexity gradually increases, certain criteria (which are called selection criteria), ass through a minimum The goal here is to resent a ractical method to realize NSRBN using blocks of the neurons function Using NSRBN network architecture, an unknown function f can be aroximated by the direct use of NSRBN of degree u to d for divide f u into blocks of equal-degree terms The inductive learning algorithm roceeds as follows We denote d an algorithmic ste at which f d is added to the network Therefore an unknown function f is successively aroximated by d = σ ( ) ( + ) (3) net o = d where weights in f k- are frozen once the k-th degree f k is added Neurobiological Plausibility of NSRBN Are NSBRN comuting structures even remotely neurobiologically lausible? In neurohysiology, the ossibility that dendritic comutations could include local multilicative nonlinearities is widely acceted Mel [6] has recently roosed that clusteron as an abstraction for a comlex neuron that can extract higher order statistics from inut stimuli In his model, a dendritic tree receives weighted synatic contacts from a set of afferent axons Each synatic contact is given by a roduct of direct stimulus intensity and a weighted sum of neighborhood activity We note that this descrition translates to an NSRBN, which can be considered as a mathematical abstraction of the clusteron model 3 The PARMA Model x, where v denotes the year, =,, ω and ω is the Let us conseder the original eriodic series v, number of time intervals in the year Assuming that the distribution of the series is skewed, an aroriated transformation can be used to transform x v, to the normal series y v, Then the eriodic PARMA model for y v, can be written as y v, = µ + σ z v, (4) where µ and σ are the eriodic mean and eriodic standard deviation and z v, may be reresented by an PARMA model The PARMA(,q) model with time-varying coefficients as q, = j, v, j θi, εv, i + εv, j = i= z v z φ (5) Proceedings of the 4th International Conference on Comutational Intelligence and Multimedia Alications (ICCIMA')

4 and where φ, θ, j and i are time varying autoregressive and moving average coefficients, resectively, ε v, is an indeendent and identically distributed normal random variable 4 Simulation Results Our exeriment uses data of some reresentative dams in oeration in Brazil[7][8] Passo Real The dam is located on the Jacuí River, situated km northwest of Porto Alegre (Brazil) and km ustream from Jacuí Hydroelectric Power Plant (historical data 93 to 997) Itumbiara The Itumbiara Dam is situated on the Paranaiba River on the boundary between the municial districts of Itumbiara in the state of Goias and Tuaciguara in Minas Gerais, aroximately 8 km northeasth of Rio de Janeiro and 3 km south of Brasilia Salto Santiago The Salto Santiago Hydroelectric Project is located on the Iguaçu River, state of Paraná, 34 km west of Curitiba, between the towns of Laranjeiras do Sul and Choinzinho (historical data 969 to 997) It is located 5 km downstream from the confluence of the Iguaçu and Areia Rivers, and 4 km from the City of Curitiba Furnas The Furnas hydroelectric lant is situated in the southeastern region of the state of Minas Gerais in the municial district of Alinóolis, on the Grande River, the largest tributary of the Paraná River (historical data 973 to997) Foz do Areia The Foz do Areia Powerlant is the most imortant roject on the Iguaçu River since it is the largest and the uer-most with regard to location on the river It controls and regulates the river discharge to the other owerlants downstream (historical data 969 to 997) Sobradinho - The Sobradinho hydroelectric lant is located in a submedium stretch of São Francisco River, Bahia State, circa 45 km on the ustream side of Juazeiro and Petrilina cities (historical data 93 to 997) In the evaluation of the erformance of the NSRBN and of the PARMA((i),q(i)) model (Periodic ARMA models) we used the absolute average ercentual error (AAPE) (Equation 9) AAPE(%) = N { [ Z Zo / Zo} N i = (6) where: Z forecast value ; Z o measured value and N number of values Table shows a comarative study of the better results obtained with Box-Jenkins models and the results obtained with neural network (NSRBN) NSRBN PARMA MODELS Hydroelectric lant AAPE AAPE Passo Real 36, 44,6 Itumbiara 6,7 7,7 Salto Santiago 3,5 33,7 Furnas,3 3,3 Foz do Areia 3, 33, Sobradinho 3, 4, Table - Comarative results of using a single NSRBN versus 5 Box-Jenkins models Proceedings of the 4th International Conference on Comutational Intelligence and Multimedia Alications (ICCIMA')

5 The Figure 3 shows, how examle, the tracking behavior of the redicted values by the NSRBN, redicted values by the 5 Box-Jenkins models for the year of 997 Passo Real Passo Real Streamflow (m3 /s) weeks Historical data PARMA Streamflow (m3 /s) weeks Historical data NSRBN Figure 3 Inflow forecast for the Passo Real Figure 4 Inflow forecast for the Passo Real Historical Data x PARMA Model (997) Historical Data x NSRBN (997) 5 Conclusions The roblem of forecasting river flows is closely related with the dificulties that sciences find in exlaining the hysical rocess of the hydrologic cycle It s necessary the use of mathematical models, though may be a rough simlification of the real roblem, to reresent this comlex system The NSRBN algorithm is a owerful modelling and rediction of comlex linear or non-linear multi-inut/multioutut systems The results obtained with this model were comared to those obtained with PARMA models The results show a more accurate redictions using the NSRBN, with a reduction in forecasting error of at least 9% NSRBN rovide good results because the well-known roblems of an otimal (subjective) choice of the neural network architecture are solved in the NSRBN algorithms by means of an adative synthesis (objective choice) of the architecture to rovide a arsimonious model for the articular desired function The statistical models in general, do not generate those good results References [] GEP Box & GM Jenkins, Time Series Analysis - Forecasting and Control Holden-Day, California, 976 [] J D Salas, J W Delleur, V Yevjevich and W L Lane Alied Modeling of Hydrologic Time Series Water Resources Publications, Colorado, 98 [3] M J S Valença Analysis and Design of the constructive neural networks for comlex systems modeling (in ortuguese) PhD These, UFPE, Brazil, 999 [4] M J S Valença and T B Ludermir Self-organization Sigmoidal Blocks Networks International Joint Conference on Neural Networks (IJCNN), IEEE, Book of Summaries (86), Washington, DC, July 999 [5] M J S Valença and T B Ludermir Self-organization Neurons Blocks Networks International Conference on Comutational Intelligence and Multimedia Alications (ICCIMA), IEEE, Acceted, New Delhi, India, Setember 999 [6] B W Mel The clusteron: Toward a simle abstraction for a comlex neuron In advances in Neural Information Processing Systems 4, S J Hanson, J E Moody, and R P Limann, Eds San Marco, CA: Morgan-Kaufmann,ages 35-4, San Marco, 99 [7] M J S Valença and T B Ludermir Multilicative-Additive Neural Networks with Active Neurons International Joint Conference on Neural Networks (IJCNN), IEEE, Book of Summaries (73), Washington, DC, July 999 [8] M J S Valença and T B Ludermir Self-organizing modeling in forecasting daily river flows V Brazilian Symosium on Neural Networks-Brazilian Comuter Society, IEEE, : -4, Belo Horizonte 998 Proceedings of the 4th International Conference on Comutational Intelligence and Multimedia Alications (ICCIMA')