NONLINEAR TIME SERIES ANALYSIS

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1 Diss ETH No NONLINEAR TIME SERIES ANALYSIS WITH CONNECTIONIST NETWORKS A dissertation submitted to the SWISS FEDERAL INSTITUTE OF TECHNOLOGY ZURICH for the degree of Doctor of Natural Sciences presented by Claas de Groot Dipl.-Phys. born May 14th, 1963 citizen of Germany accepted on the recommendation of PD Dr. D. Wiirtz, examiner Prof. Dr. K. Hepp, co-examiner Prof. Dr. H. R. Kiinsch, co-examiner 1993

2 Abstract We investigate connectionist networks for nonlinear time series analysis Oui ex penments were carried out with single hidden layer feedforward networks with hyperbolic tangent at the hidden units and linear transfer functions at input and output units We demonstrate the need for a complete method which allows mean ingful analysis from raw data to the final time series model This method is based on pragmatic reasoning rather than on statistical theory It aims at the practi tioner For this reason a software package has been developed which implements the method efficiently In multivariate modeling we perform nonparametric tests in order to identify indicators and eliminate "meaningless" data from the sample After pattern for mation we propose a clustering scheme m order to determine the members of the training and validation set Data involved in the analysis is standardized We make use of the linear potential of the network m computing a complete initial linear so lution This substitutes the common random initialization of net parameters This procedure requires introduction of an additional parameter, which controls the va lidity of the linearization The complete set of weights is obtained from principal component analysis (PCA) of input data as well as Box Jenkins procedure PCA additionally allows a rational specification of the number of hidden units for the network topology The estimation of parameters starts from the linear solution A standard optimization algorithm of second order is used for this purpose During optimization, characteristic values of the network are monitored These values are the performance criterion, the eigenvalues of the correlation matrix of hidden umts outputs, smgulax values of the weights at the hidden layer, and the slope of the regression comparing the linear and the nonlinear network Since the initial weight estimate has known properties, changes m the monitored values during optimiza tion can be interpreted (at least) qualitatively This interpretation addresses the following problems The network has to make use of its nonhnearity, otherwise linear modeling would be adequate, the slope of the regression helps answering in

3 this question. If there are more hidden nodes than just one operating in the linear approximation, the linearity can be assigned to a single hidden node. This can be detected with the help of singular value decomposition. Linear correlations between hidden nodes indicate a misspecification of the hidden layer. The eigenvalues of the correlation matrix inform about this situation. Monitoring of characteristic values allows a rational basis for decisions about important features of the connectionist solution of the time series problem. Iterations and meaningful improvements of the model are facilitated and supported by this technique. We also propose "antithetic" combination of models. This may result in a reduction of the forecasting error due to weakly correlated prediction errors. We demonstrate our connectionist network method for time series analysis by applying it to case studies of real life data: Zurich sunspots data, West Bohemian power consumption, and currency exchange rates. We compared our method to linear models and networks with randomly initialized weights. Out method achieves models which were better than "random" nets, and the final model was better than the linear model. IV

4 Kurzfassung Gegenstand unserer Untersuchungen ist die nichtlineare Zeitreihenanalyse mit konnektionistischen Netzen. Der von uns gcwahlte Netztyp ist das Netzwerk mit Vorwartsstruktur und einer verborgenen Schicht. Wahrend die Knoten der Eingangsund Ausgangsschicht eine lineare Transferfunktion besitzen, ist die Transferfunktion der verborgenen Schicht der Tangens hyperbolicus. Einleitend begrunden wir den von uns gewahlten "vereinheitlichenden" Ansatz, da eine rationale Analyse der vorgelegten Zeitreihe ein Vorgehen mit aufeinander abgestimmten Teclmiken erzwingt. Unsere Methode ist fur den Praktiker gedacht; sie ist in einem effizienten Programmpaket verwirklicht, das den Praktiker iiber alle Schritte der Analyse auf dem Computer verfugen lasst. Fur die multivariate Analyse fuhren wir nichtparametrische Tests durch mit dem Ziel, Indikatoren zu finden und die Eingangsdaten auf moglichst grossen Informationsgehalt hin auszuwahlen. Aus diesen Daten werden die Muster gebildet, die wiederum mit einem eigenen Algorithmus in zwei Gruppen eingeteilt werden, namlich die Menge der Lernbeispiele und der Vahdierungsbeispiele. Wir standardisieren die Daten fur die Analyse. Das lineare Potential des Netzwerks nutzen wir aus, indem wir eine vollstandige Startlosung aus den Mustern berechnen, statt die Anfangsgewichte zufallig festzulegen. Dies erfordert einen Parameter, der die Giiltigkeit der linearen Approximation bestimmt. Die Startlosung erhalt man aus der Faktorenanalyse und dem Box-Jenkins Ansatz. Die Faktorenanalyse erlaubt zugleich eine begriindete Festlegung der Anzahl an verborgenen Knoten. Die Schatzung der Gewichte erhalt die berechnete Losung als Startwert. Wir verwenden einen Optimierungsalgorithmus zweiter Ordnung. Wahrend der Optimierung zeichnen wir charakteristische Werte auf, mit denen sich Aussagen iiber die Qualitat der Netzwerklosung treffen lassen. Diese Werte umfassen den Wert der Zielfunktion, Eigenwerte der Korrelationsmatrix der Ausgange der verborgenen Knoten, Steigung der Regression zwischen linearem und nichtlinearem Netz und Singularwertzerlegung der Gewichte in der verborgenen Schicht. Mit Hilfe dieser v

5 Aufzeiclmung lassen sicli folgende Probleme qualitativ eihellen: Die Regression erlaubt eine Aussage, ob das Netz von seinen Nichtlinearitaten Gebrauch maclit. Falls es im linearen Bereich arbeitet, kann auf die konnektionistische Methode verzichtet werden. Falls mehr als ein Knoten der verborgenen Schicht im linearen Bereich ar beitet, kann diese Linearitiit in einem einzigen Knoten zusammengezogen werden: dariiber informiert die Singularwertzerlegung. Falls lineare Korrelationen zwischen Knoten der verborgenen Schicht bestehen, deutet dies hin auf zu viele Knoten in dieser Schicht, daher die Eigenwertanalyse der verborgenen Knoten. Da die Aufzeichnung dieser charakteristischen Werte Verbesserungen des Netzmodells auf einer rationalen Grundlage ermoglicht, wird das Modellieren insgesamt erleichtert. Eine "antithetische" Kombination von verschiedenen Modellen fuhrt schliesshch zu einem weiter verbesserten Modell, falls die Fehler der Vorhersage wenig korreliert sind. Wir wenden die von uns entwickelte Methode der nichtlinearen Zeitreihenanalyse mit konnektionistischen Netzen in Fallstudien an: die Zuricher Reihe der Sonnenflecken, Stromverbrauchsdaten aus Westbohmen und Wechselkurse. Wir stellen zu Vergleichszwecken auch ein lineares Modell auf sowie ein konnektionistisches Modell, das mit zufallig gewahlten Gewichten initialisiert wurde. Unsere Methode fuhrt auf bessere Resultate als die aus dem "zufalligen" Modell, das endgultige Modell ist jeweils auch besser als das lineare Vergleichsmodell. VI

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