### Abstract

In Principal Component Analysis (PCA), the dimension of the signal subspace is detected by counting the number of eigenvalues of a covariance matrix that are above a threshold. Random matrix theory provides accurate estimates for this threshold if the underlying data matrix has independent identically distributed columns. However, in time series analysis, the underlying data matrix has a Hankel or Toeplitz structure, and the columns are not independent. Using an empirical approach, we observe that the largest eigenvalue is fitted well by a Generalized Extreme Value (GEV) distribution, and we obtain accurate estimates for the thresholds to be used in a sequential rank detection test. In contrast to AIC or MDL, this provides a parameter that controls the probability of false alarm. Also a lower bound is presented for the rank detection rate of threshold-based detection for rank-1 problems.

Original language | English |
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Title of host publication | 28th European Signal Processing Conference (EUSIPCO 2020) |

Place of Publication | Amsterdam (Netherlands) |

Publisher | Eurasip |

Pages | 1911-1915 |

Number of pages | 5 |

ISBN (Electronic) | 978-9-0827-9705-3 |

Publication status | Published - 1 Aug 2020 |

Event | EUSIPCO 2020: The 28th European Signal Processing Conference - Amsterdam, Netherlands Duration: 18 Jan 2021 → 22 Jan 2021 Conference number: 28th |

### Conference

Conference | EUSIPCO 2020 |
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Country | Netherlands |

City | Amsterdam |

Period | 18/01/21 → 22/01/21 |

Other | Date change due to COVID-19 (former date August 24-28 2020) |

### Keywords

- PCA
- structured Wishart matrix
- rank detection
- Generalized Extreme Value

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## Cite this

van der Veen, A. J., Romme, J. P. A., & Cui, Y. (2020). Rank detection thresholds for Hankel or Toeplitz data matrices. In

*28th European Signal Processing Conference (EUSIPCO 2020)*(pp. 1911-1915). Eurasip.