Analysis and Forecasting of Time Series by Averaged Scalar Products of Flow Vectors

The relat ionship between the quality of state space reconst ruction and the accuracy in time series forecast ing is analyzed. The averaged scalar product of the dynamical system flowvectors has been used to give a degree of determinism to the selected state space reconst ruct ion. This value helps dist inguish between those regions of the state space where predictions will be accurate and those where they are not . A time series measured in an industri al environment where noise is present is used as an example. It is shown that prediction methods used to estimate futu re values play a less important role than a good reconstruction of the state space itself. 1. Int roduct ion Much work has recently been done in nonlinear time series pr edictio n [1-4] . However , most of this effort has been focused on t ime series originat ing from dyn amical systems where there is a well-defined, alt hough complex and ofte n analytic, underlying model. For such systems, a great number of different techniques have been developed to tackle the pr edict ion problem. These techn iques include state space reconst ru ction of chaotic sys tems [57] as well as var ious methods to ap proximate future values from eit her local [2, 8] or global models [1,9]. State space reconstruction is aimed at obtaining a trajectory in t he state space leading to a determinist ic reconstru cti on of the t ime series. This is usually accomplished using Takens' theorem [10], which states that if the underlying dynam ical syst em has dimension d, t he reconstru ct ion , usually called 26 C. Santa Cruz, R. Huerta, J. R. Dorronsoro, and Vicente Lopez an embedding, can be carr ied out using delay vecto rs in an m-dimensional space (m 2': 2d + 1). Unlike the behav ior of irregular time series, the state space usua lly demonst rates simplicity and regulari ty. Vector fields of the state space are approximated to est ima te future values. The quality of the predictions st rongly depend s on the quality of the state space reconstruction. Furthermore, any lack of accuracy or defect in t he state space reconstruction has consequences for t he prediction of t ime series. Therefore, a measure of the quality of a state space reconst ruction gives an est ima te of the goodness of the forecasting model. The average d scalar product P of the dynami cal system flow vecto rs [5] has recently been used to determine optimal state space reconstruction by maximizing the P -value as a function of both the state space dimension and the time delay. The P-value can be used to qualify th e "degree of determinism" of state space reconstruction ; tha t is, it provides a measure of how para llel the local flow vectors are throughout th e state space. It has been shown [5] th at for tim e series originating from ordinary differential equa tions the P-value can be raised close to unity. This value indic ates that the neighboring flow vectors in th e reconstructed state space are para llel to each other. On t he oth er hand, neighboring flows for time series contamina ted wit h noise or originati ng from ill-defined syste ms will be far from par allel. This leads to lower P-values. This means that th e syste ms are less determini stic than the former, since the flow of point s along the state space are not so well-defined. Hence, the predict ion will not be as accurate . This work highlight s the exist ing relati onship between the characteristics of state space reconstruction of t ime series and t he quality of its predictions in light of results derived from the P-value. Some of th e conclusions drawn from state space analysis apply to both , substant iat ing t he results obtained in the pr ediction and devising new prediction algorithms. As an example, we will use the state space reconstruction and time series prediction of a variable measured in an industrial environment . This t ime series corresponds to an imp ortant temp erature measur ed in a petrochemical plant. Forecasting is one of the techniques that has been int egrat ed into th e HINT proj ect (ESP RIT) for int elligent control. The series is sampled every five minutes and the time histor y over three month s (25,000 dat a points) is used. Figure 1 illustrates 20 hour s of the act ua l time series and the filtered signal obtained after removing th e high frequences using a low-pass filter . The filtered signal will be used throughou t this paper . Since the time series is measur ed in an industri al setting where a grea t numb er of uncont rolled factor s are no doub t present , the und erlying mod el is not well-defined. This lack of accuracy in the definit ion of the system leads to various problems in both state space reconstructi on and the quality of the prediction that would ot herwise not appear in sit uations where a well-defined model existed . The article is st ructured as follows. Section 2 describ es problems th at arise when performing state space reconstructions of real tim e series. Section 3 ana lyses t he forecasting of t ime series using result s obtained in the previous section, and finally, section 4 summarizes the conclusions of this pap er . Analysis and Forecasting of Tim e Series 10 ,------.----,--------,----------,