686 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 45, NO. 3, MARCH 1997 Model Selection through a Statistical Analysis of the Global Minimum of a Weighted Nonlinear Least Squares Cost Function Rik Pintelon, Johan Schoukens, and Gerd Vandersteen Abstract—This paper presents a model selection algorithm for the identification of parametric models that are linear in the measurements. It is based on the mean and variance expressions of the global minimum of a weighted nonlinear least squares cost function. The method requires the knowledge of the noise covariance matrix but does not assume that the true model belongs to the model set. Unlike the traditional order estimation methods available in literature, the presented technique allows to detect undermodeling. The theory is illustrated by simulations on signal modeling and system identification problems and by one real measurement example. I. INTRODUCTION I N SYSTEM identification and signal modeling, a lot of research effort is spent to the identification of paramet- ric models. An identification procedure typically consists of applying iterative model selection and parameter estimation. The model selection (order estimation) is still the most critical step in the identification process. In particular, the detection of undermodeling 1 is a major problem for the model selection methods available in literature. Most of the classical model selection algorithms like, for example, the Akaike information criterion (AIC), the -test, and the minimum description length (MDL) assume implicitly or explicitly that the true model belongs to the model set [1]–[5]. The shortcomings of these traditional methods have been extensively studied in [6]. Some recent methods [7] and [8] take into account the model errors but still are not able to detect undermodeling [6]. Most of the classical methods are, however, well suited to detect overmodeling. 2 Some model validation tests reveal a certain class of model errors. For example, statistical tests of whiteness of the residuals [4] may detect unmodeled dynamics in system identification problems but are insensitive to model Manuscript received July 14, 1995; revised August 15, 1996. This work was supported by the National Fund for Scientific Research of Belgium, the Flemish Government (GOA-IMMI), and the Belgian Programme on Interuniveristy Poles of Attraction initiated by the Belgian State, Prime Minister’s Office, Science Policy programming (IUAP 2). The associate editor coordinating the review of this paper and approving it for publication was Dr. Zhi Ding. The authors are with the Department of Electrical Measurements, Vrije Universiteit Brussel, 1050 Brussels, Belgium. Publisher Item Identifier S 1053-587X(97)01887-4. 1 Undermodeling is, for example, due to unmodeled dynamics and/or nonlinear distortions in system identification or, for example, due to too small a number of sinewaves and/or nonperiodic deterministic disturbances in signal modeling. 2 Overmodeling occurs if the considered model set includes the true model and if it is described by too many parameters. errors like nonlinear distortions [9], which behave as white noise in the residuals. The main reason why the model selection and model validation algorithms fail to detect (all possible causes of) undermodeling is that their stochastic behavior in the presence of model errors is unknown. To reveal this behavior, the second-order properties of the disturbing noise are required. Another approach, which has been shown to be successful for NARMAX models, consists of analyzing fourth-order correlations of the residuals [6], [10]. This paper presents an order estimation algorithm that detects undermodeling for a general class of models. The al- gorithm is based on expressions for the mean and the variance of the global minimum of a weighted nonlinear least squares cost function. It is a generalization of the results obtained in [11] and [12] that were derived for a particular model and are valid for independent Gaussian distributed errors only. The prediction of the stochastic behavior of the cost function requires the knowledge of the mean, the covariance matrix, and the kurtosis factor of the disturbing noise. This knowledge can be obtained from repeated experiments or from noise (cross-) power measurements when no excitation signal is present (see, for example, [13] for frequency domain identification). It is, however, not necessary to assume that the true model is included in the considered model set. The paper is organized as follows. Sections II and III define, respectively, the class of parametric models and the class of weighted nonlinear least squares estimators for which the theory applies. The expected value and the variance of the global minimum of the cost function are calculated in Section IV for Gaussian and non-Gaussian errors. The model selection algorithm is described in Section V, while Section VI contains some simulation examples. II. THE SEMILINEAR MODEL Consider the following general model based on obser- vations (1) which is linear-in-the-measurements and (non-) linear-in-the-model-parameters , and where and . , , and are fixed integers independent of the number of observations . Each time a new observation is added, the number of model equations and the number of measurements increase 1053–587X/97$10.00  1997 IEEE