The asymptotic local approach to change detection and model validation.

*(English)*Zbl 0628.93071This paper introduces a general procedure for approaching the following three related problems.

A) Change detection. Let \(\theta_*\) denote the unknown parameter vector describing the system under study; and let \(\theta_ 0\) denote a nominal model available to the user. Then the problem is to test the following hypotheses: \[ H_ 0: \theta_*=\theta_ 0\text{ (i.e no change)} \]

\[ H_ 1: \theta_*= \begin{cases} \theta_ 0 &\text{for \(1\leq t<\alpha N,\)} \\ \theta_ 0+\theta/\sqrt{N} &\text{for \(\alpha N\leq t\leq N,\)}\end{cases} \] where N is the number of data points, \(\alpha\in [0,1]\), and \(\theta\neq 0\) is an unknown vector. If \(H_ 1\) is selected then it may also be required to determine the moment of change \(\alpha N.\)

B) Model validation. In this case, the null and the alternative hypotheses are the following \[ H_ 0: \theta_*= \theta_ 0\text{ (the model is valid), } H_ 1: \theta_*= \theta_ 0+ \theta/\sqrt{N}. \] Thus problem B is a special case of a (just set \(\alpha =0\) in the statement of A).

C) Change diagnosis. Assuming that \(H_ 1\) is the selected hypothesis, the problem is to decide, what parameters of the parameter vector have changed and how. This problem can also be formulated as a special form of change detection for which the hypotheses are \[ H_ 0: \theta_*= \theta_ 0,\quad H_ 1: \theta_*= \theta_ 0+ \theta/\sqrt{N}, \] where \(\theta\neq 0\) belongs to a subset of the parameter space.

The basic idea of the procedure proposed for solving the above problems can be explained as follows. A properly selected random vector is shown to have a zero-mean Gaussian distribution under \(H_ 0\) and a Gaussian distribution with the same covariance matrix but a nonzero mean under \(H_ 1\delta\). Thus the problem of deciding between \(H_ 0\) and \(H_ 1\) is reduced to that of detecting a change in the mean of independent Gaussian vector random variables.

The distributional results in the paper are derived under fairly general conditions which allow the presence of nonstationary nuissance parameters. This is a considerable extension of the previously available theory.

In closing this review we mention an aspect which appears to require further study. In practice \(\theta_ 0\) will often be a vector of estimated parameters. This means that even under \(H_ 0\) (the hypothesis of no change) \(\theta_ 0\) will differ from \(\theta_*\). This difference (which is \(O(1/\sqrt{N})\) under quite general conditions) may affect the (asymptotic) distributional results derived in the paper as suggests the recent work by T. Söderström and the reviewer, “On covariance function tests used in system identification”, UPTEC Report, Uppsala University, Sweden, 1987.

A) Change detection. Let \(\theta_*\) denote the unknown parameter vector describing the system under study; and let \(\theta_ 0\) denote a nominal model available to the user. Then the problem is to test the following hypotheses: \[ H_ 0: \theta_*=\theta_ 0\text{ (i.e no change)} \]

\[ H_ 1: \theta_*= \begin{cases} \theta_ 0 &\text{for \(1\leq t<\alpha N,\)} \\ \theta_ 0+\theta/\sqrt{N} &\text{for \(\alpha N\leq t\leq N,\)}\end{cases} \] where N is the number of data points, \(\alpha\in [0,1]\), and \(\theta\neq 0\) is an unknown vector. If \(H_ 1\) is selected then it may also be required to determine the moment of change \(\alpha N.\)

B) Model validation. In this case, the null and the alternative hypotheses are the following \[ H_ 0: \theta_*= \theta_ 0\text{ (the model is valid), } H_ 1: \theta_*= \theta_ 0+ \theta/\sqrt{N}. \] Thus problem B is a special case of a (just set \(\alpha =0\) in the statement of A).

C) Change diagnosis. Assuming that \(H_ 1\) is the selected hypothesis, the problem is to decide, what parameters of the parameter vector have changed and how. This problem can also be formulated as a special form of change detection for which the hypotheses are \[ H_ 0: \theta_*= \theta_ 0,\quad H_ 1: \theta_*= \theta_ 0+ \theta/\sqrt{N}, \] where \(\theta\neq 0\) belongs to a subset of the parameter space.

The basic idea of the procedure proposed for solving the above problems can be explained as follows. A properly selected random vector is shown to have a zero-mean Gaussian distribution under \(H_ 0\) and a Gaussian distribution with the same covariance matrix but a nonzero mean under \(H_ 1\delta\). Thus the problem of deciding between \(H_ 0\) and \(H_ 1\) is reduced to that of detecting a change in the mean of independent Gaussian vector random variables.

The distributional results in the paper are derived under fairly general conditions which allow the presence of nonstationary nuissance parameters. This is a considerable extension of the previously available theory.

In closing this review we mention an aspect which appears to require further study. In practice \(\theta_ 0\) will often be a vector of estimated parameters. This means that even under \(H_ 0\) (the hypothesis of no change) \(\theta_ 0\) will differ from \(\theta_*\). This difference (which is \(O(1/\sqrt{N})\) under quite general conditions) may affect the (asymptotic) distributional results derived in the paper as suggests the recent work by T. Söderström and the reviewer, “On covariance function tests used in system identification”, UPTEC Report, Uppsala University, Sweden, 1987.

Reviewer: P.Stoica

##### MSC:

93E12 | Identification in stochastic control theory |

62F03 | Parametric hypothesis testing |

93E10 | Estimation and detection in stochastic control theory |

60G35 | Signal detection and filtering (aspects of stochastic processes) |

62E20 | Asymptotic distribution theory in statistics |

93E25 | Computational methods in stochastic control (MSC2010) |