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  • Multivariate statistical models for seasonal climate prediction and

    climate downscaling by

    Alex Jason Cannon

    B.Sc., The University of British Columbia, 1995 Dipl. Meteorology, The University of British Columbia, 1996

    M.Sc., The University of British Columbia, 2000

    A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF

    Doctor of Philosophy

    in

    The Faculty of Graduate Studies

    (Atmospheric Science)

    The University Of British Columbia

    December, 2008

    c© Alex Jason Cannon 2008

  • Abstract

    This dissertation develops multivariate statistical models for seasonal forecasting and down-

    scaling of climate variables. In the case of seasonal climate forecasting, where record lengths

    are typically short and signal-to-noise ratios are low, particularly at long lead-times, forecast

    models must be robust against noise. To this end, two models are developed. Robust nonlin-

    ear canonical correlation analysis, which introduces robust cost functions to an existing model

    architecture, is outlined in Chapter 2. Nonlinear principal predictor analysis, the nonlinear

    extension of principal predictor analysis, a linear model of intermediate complexity between

    multivariate regression and canonical correlation analysis, is developed in Chapter 3. In the

    case of climate downscaling, the goal is to predict values of weather elements observed at local

    or regional scales from the synoptic-scale atmospheric circulation, usually for the purpose of

    generating climate scenarios from Global Climate Models. In this context, models must not

    only be accurate in terms of traditional model verification statistics, but they must also be

    able to replicate statistical properties of the historical observations. When downscaling series

    observed at multiple sites, correctly specifying relationships between sites is of key concern.

    Three models are developed for multi-site downscaling. Chapter 4 introduces nonlinear analog

    predictor analysis, a hybrid model that couples a neural network to an analog model. The

    neural network maps the original predictors to a lower-dimensional space such that predictions

    from the analog model are improved. Multivariate ridge regression with negative values of the

    ridge parameters is introduced in Chapter 5 as a means of performing expanded downscaling,

    which is a linear model that constrains the covariance matrix of model predictions to match

    that of observations. The expanded Bernoulli-gamma density network, a nonlinear probabilis-

    tic extension of expanded downscaling, is introducted in Chapter 6 for multi-site precipitation

    downscaling. The single-site model is extended by allowing multiple predictands and by adopt-

    ing the expanded downscaling covariance constraint.

    ii

  • Table of Contents

    Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

    Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

    List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

    List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

    Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv

    Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

    Statement of Co-Authorship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi

    1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

    1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

    1.2 Linearity and nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

    1.3 Linear multivariate statistical models . . . . . . . . . . . . . . . . . . . . . . . . 3

    1.3.1 Multivariate regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

    1.3.2 Canonical correlation analysis and maximum covariance

    analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

    1.3.3 Principal predictor analysis and redundancy analysis . . . . . . . . . . . 4

    1.4 Nonlinear multivariate statistical models . . . . . . . . . . . . . . . . . . . . . . 6

    1.4.1 General methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

    1.4.2 Nonlinear multivariate regression . . . . . . . . . . . . . . . . . . . . . . 6

    1.4.3 Nonlinear canonical correlation analysis . . . . . . . . . . . . . . . . . . 7

    1.5 Seasonal climate prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

    1.5.1 Linear methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

    1.5.2 Nonlinear methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

    1.6 Climate downscaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

    1.6.1 Linear methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

    1.6.2 Nonlinear methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

    1.6.3 Hybrid methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

    1.7 Research goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

    iii

  • Table of Contents

    Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

    2 Robust nonlinear canonical correlation analysis . . . . . . . . . . . . . . . . . . 21

    2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

    2.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

    2.2.1 NLCCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

    2.2.2 Biweight midcorrelation . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

    2.2.3 Lp-norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

    2.2.4 Robust NLCCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

    2.3 Synthetic test problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

    2.3.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

    2.3.2 Training and testing procedure . . . . . . . . . . . . . . . . . . . . . . . . 31

    2.3.3 Model performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

    2.4 Seasonal prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

    2.4.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

    2.4.2 Training and testing procedure . . . . . . . . . . . . . . . . . . . . . . . . 33

    2.4.3 Skill for models with one mode . . . . . . . . . . . . . . . . . . . . . . . . 35

    2.4.4 Skill for models with two modes . . . . . . . . . . . . . . . . . . . . . . . 41

    2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

    Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

    3 Nonlinear principal predictor analysis . . . . . . . . . . . . . . . . . . . . . . . . 46

    3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

    3.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

    3.3 Application to the Lorenz system . . . . . . . . . . . . . . . . . . . . . . . . . . 54

    3.4 Discussion and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

    Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

    4 Nonlinear analog predictor analysis . . . . . . . . . . . . . . . . . . . . . . . . . 67

    4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

    4.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

    4.2.1 Analog model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

    4.2.2 Nonlinear Principal Predictor Analysis . . . . . . . . . . . . . . . . . . . 70

    4.2.3 Nonlinear Analog Predictor Analysis . . . . . . . . . . . . . . . . . . . . 72

    4.3 Synthetic benchmark tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

    4.3.1 Linear benchmark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

    4.3.2 Nonlinear benchmark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

    4.3.3 Multicollinear benchmark . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

    4.3.4 Multiplicative benchmark . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

    iv

  • Table of Contents

    4.3.5 Synthetic benchmark results . . . . . . . . . . . . . . . . . . . . . . . . . 74

    4.4 Multivariate downscaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

    4.4.1 Synoptic-scale atmospheric circulation dataset . . . . . . . . . . . . . . . 77

    4.4.2 Whistler temperature and precipitation dataset . . . . . . . . . . . . . . 77

    4.4.3 Multi-site precipitation dataset . . . . . . . . . . . . . . . . . . . . . . . . 81

    4.5 Discussion and conclusion . . . . . . . . .

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