### Abstract

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

Place of Publication | Birmingham |

Publisher | Aston University |

Number of pages | 20 |

ISBN (Print) | NCRG/97/011 |

Publication status | Published - 3 Jul 1997 |

### Fingerprint

### Keywords

- regression
- Gaussian assumptions
- Bayesian prediction
- regularization
- smoothing
- posterior mean
- linear model
- infinite dimensional analogue
- principal component analysis
- Hilbert space methods
- practical statistics

### Cite this

*Gaussian regression and optimal finite dimensional linear models*. Birmingham: Aston University.

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**Gaussian regression and optimal finite dimensional linear models.** / Zhu, Huaiyu; Williams, Christopher K. I.; Rohwer, Richard; Morciniec, Michal.

Research output: Working paper › Technical report

TY - UNPB

T1 - Gaussian regression and optimal finite dimensional linear models

AU - Zhu, Huaiyu

AU - Williams, Christopher K. I.

AU - Rohwer, Richard

AU - Morciniec, Michal

PY - 1997/7/3

Y1 - 1997/7/3

N2 - The problem of regression under Gaussian assumptions is treated generally. The relationship between Bayesian prediction, regularization and smoothing is elucidated. The ideal regression is the posterior mean and its computation scales as O(n3), where n is the sample size. We show that the optimal m-dimensional linear model under a given prior is spanned by the first m eigenfunctions of a covariance operator, which is a trace-class operator. This is an infinite dimensional analogue of principal component analysis. The importance of Hilbert space methods to practical statistics is also discussed.

AB - The problem of regression under Gaussian assumptions is treated generally. The relationship between Bayesian prediction, regularization and smoothing is elucidated. The ideal regression is the posterior mean and its computation scales as O(n3), where n is the sample size. We show that the optimal m-dimensional linear model under a given prior is spanned by the first m eigenfunctions of a covariance operator, which is a trace-class operator. This is an infinite dimensional analogue of principal component analysis. The importance of Hilbert space methods to practical statistics is also discussed.

KW - regression

KW - Gaussian assumptions

KW - Bayesian prediction

KW - regularization

KW - smoothing

KW - posterior mean

KW - linear model

KW - infinite dimensional analogue

KW - principal component analysis

KW - Hilbert space methods

KW - practical statistics

M3 - Technical report

SN - NCRG/97/011

BT - Gaussian regression and optimal finite dimensional linear models

PB - Aston University

CY - Birmingham

ER -