Searching for Dynamical Resemblance Between Time Series - A Method Based on Nonlinear AutoRegressive Models Gladstone Barbosa Alves ¥ Viviane Cota Silva ¥ Marcelo Vieira Corrêa § ¥ Laboratório de Modelagem, Análise e Controle de Sistemas Não-Lineares Centro de Pesquisa e Desenvolvimento em Engenharia Elétrica – CPDEE Universidade Federal de Minas Gerais – UFMG Av. Antônio Carlos 6627, 31270-901 Belo Horizonte, M.G., Brasil § Centro Universitário do Leste de Minas Gerais – UnilesteMG Av. Tancredo Neves, 3500, 35170-056 Coronel Fabriciano, M.G., Brasil [email_address] , vivianne@cpdee.ufmg.br, [email_address] XIV Congresso Brasileiro de Automática Natal/RN 02-05/setembro, 2002

Summary Introduction; Background: the polynomial NARMAX representation; recovering the eigenvalue function from polynomial NARX models; The procedure to search for dynamical resemblance based on polynomial NARMAX models; Example: data from a monthly average river flow; Concluding remarks.

Introduction;

Background:

the polynomial NARMAX representation;

recovering the eigenvalue function from polynomial NARX models;

The procedure to search for dynamical resemblance based on polynomial NARMAX models;

Example: data from a monthly average river flow;

Concluding remarks.

Introduction  Methods for time series analysis:  linear systems theory;  nonlinear dynamics theory. NARMAX:  a suggestion for the nonstationary and short time series analysis problem;  an advantageous tool to detect dynamical closeness.

 Methods for time series analysis:

 linear systems theory;

 nonlinear dynamics theory.

NARMAX:

 a suggestion for the nonstationary and short time series analysis problem;

 an advantageous tool to detect dynamical closeness.

Background The polynomial NARMAX model representation First-order NARX model

Background Recovering the eigenvalue function from a first-order polynomial NARX model: linearization For models with the eigenvalues will be determined as the roots of a polynomial with degree

The procedure to search for dynamical resemblance based on polynomial NARX models  Definition of a model input: correlated time series/a sine wave with period similar to that present in the time series;  Structure selection(using the entire data set) of the NARMAX model which will be kept fixed during all the procedure: ERR;  Selection of appropriate data windows to be used in the parameter estimation of the chosen model structure: prior knowledge;  Parameter estimation of the selected structure: a single model for each data window;  Determination of the eigenvalues variation of the estimated models and comparison of the recovered configurations. Steps:

Application: a case study The data: monthly average river flow Monthly average flow of Doce river at the State of Minas Gerais. Horizontal axis is the monthly samples from 1939 to 1989 and vertical axis is the average flow in mm/s (annual period). Linear autocorrelation function of the original flow data, for the first 200 lags. The lags are measured in months. The dashed lines represent a confidence interval. The signal is auto-correlated with periodicity of approximately twelve samples (annual period).

 Investigation based on linear autoregressive models: assessing dynamical properties using forecasts  Linear models were obtained using the window 1;  Forecasts of 1, 2 and 3 steps ahead over all data windows;  AR models;  AR-MA models;  It seems that the dynamics of window 1 is very similar to that of the window 6. Application: a case study  Time series empirically divided into 6 windows of 100 samples each;

 Investigation based on linear autoregressive models: assessing dynamical properties using forecasts

 Linear models were obtained using the window 1;

 Forecasts of 1, 2 and 3 steps ahead over all data windows;

 AR models;

 AR-MA models;

 It seems that the dynamics of window 1 is very similar to that of the window 6.

Analysis of the AR and AR-MA models performance along 6 windows of data Results of the linear models’ one step ahead forecasts Application: a case study

Analysis of the AR and AR-MA models performance along 6 windows of data

Results of the linear models’ two step ahead forecasts Results of the linear models’ three step ahead forecasts Application: a case study

Investigation based on polynomial NARMAX models  Input: a 12-samples period sine wave signal, with unitary amplitude;  NARX model structure selected by the ERR: Application: a case study

Investigation based on polynomial NARMAX models

 Input: a 12-samples period sine wave signal, with unitary amplitude;

 NARX model structure selected by the ERR:

Eigenvalues associated with the “stationary states” of the models obtained for each selected data window . From left to right and from top to bottom Window 1 - Window 6. Application: a case study

Concluding Remarks  Case study results:  U nderlying dynamics of windows 1 and 6 are somewhat closed (prelimary conclusion obtained by the linear model forecasts and by proposed NARMAX approach);  Suggests the existence of a “hidden” periodic cycle, implicit in the data (different of the annual one).  Proposed scheme:  It is believed that the NARMAX approach suggested here has a more consistent dynamical character than just using forecasts.

 Case study results:

 U nderlying dynamics of windows 1 and 6 are somewhat closed (prelimary conclusion obtained by the linear model forecasts and by proposed NARMAX approach);

 Suggests the existence of a “hidden” periodic cycle, implicit in the data (different of the annual one).

 Proposed scheme:

 It is believed that the NARMAX approach suggested here has a more consistent dynamical character than just using forecasts.