### Abstract

Original language | English |
---|---|

Pages (from-to) | 329-348 |

Number of pages | 19 |

Journal | Journal of Statistical Computation and Simulation |

Volume | 77 |

Issue number | 4 |

Publication status | Published - 4 Jan 2007 |

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### Keywords

- Gaussian process
- Logarithm of matrix determinant
- Power-series expansion
- Compensation

### Cite this

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*Journal of Statistical Computation and Simulation*, vol. 77, no. 4, pp. 329-348.

**Approximate implementation of logarithm of the matrix determinant in Gaussian process regression.** / Zhang, Y.; Leithead, W.E.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Approximate implementation of logarithm of the matrix determinant in Gaussian process regression

AU - Zhang, Y.

AU - Leithead, W.E.

PY - 2007/1/4

Y1 - 2007/1/4

N2 - Maximum likelihood estimation of hyperparameters in Gaussian processes (GPs) as well as other spatial regression models usually requires the evaluation of the logarithm of the matrix determinant, in short, log det. When using matrix decomposition techniques, the exact implementation of log det is of O(N3) operations, where N is the matrix dimension. In this paper, a power-series expansion-based framework is presented for approximating the log det of general positive-definite matrices. Three novel compensation schemes are proposed to further improve the approximation accuracy and computational efficiency. The proposed log det approximation requires only 50N2 operations. The theoretical analysis is substantiated by a large number of numerical experiments, including tests on randomly generated positive-definite matrices, randomly generated covariance matrices, and sequences of covariance matrices generated online in two GP regression examples. The average approximation error is ∼9%.

AB - Maximum likelihood estimation of hyperparameters in Gaussian processes (GPs) as well as other spatial regression models usually requires the evaluation of the logarithm of the matrix determinant, in short, log det. When using matrix decomposition techniques, the exact implementation of log det is of O(N3) operations, where N is the matrix dimension. In this paper, a power-series expansion-based framework is presented for approximating the log det of general positive-definite matrices. Three novel compensation schemes are proposed to further improve the approximation accuracy and computational efficiency. The proposed log det approximation requires only 50N2 operations. The theoretical analysis is substantiated by a large number of numerical experiments, including tests on randomly generated positive-definite matrices, randomly generated covariance matrices, and sequences of covariance matrices generated online in two GP regression examples. The average approximation error is ∼9%.

KW - Gaussian process

KW - Logarithm of matrix determinant

KW - Power-series expansion

KW - Compensation

M3 - Article

VL - 77

SP - 329

EP - 348

JO - Journal of Statistical Computation and Simulation

JF - Journal of Statistical Computation and Simulation

SN - 0094-9655

IS - 4

ER -