Various multivariate statistical methods have been established and proven useful for representing linear relationships in datasets. Popular methods include Canonical Correlation Analysis (CCA), Maximum Covariance Analysis (MCA), and Redundancy Analysis (RDA) that are optimal at maximizing squared correlation, squared covariance, and variance explained, respectively. Considering that such measures of relation vary only in the representation of amplitude, they may be generalized into one particular generic form introduced here as scaled covariance. Scaled covariance is equivalent to covariance between scaled data, with the scaling transformation acting to apply a specified degree of normalization and decorrelation. For intermediate scaling values, scaling may be referred to as partial whitening, a transformation that may provide a new and highly useful noise pre-filter for covariance.

Extending to multiple dimensions, Scaled Singular Value Decomposition Analysis (SSVD) provides a continuum of methods linking CCA, MCA and RDA that maximizes sample squared scaled covariance. SSVD is derived and its properties are discussed. In order to understand the behavior of SSVD in terms of its ability to isolate coupled signals as a function of scaling values, SSVD is applied to randomly-generated synthetic datasets with known coupled signals embedded in background noise. From this, it is demonstrated that intermediate SSVD solutions tend to yield higher cross-validated measures of relation in contrast to their counterparts that are optimal in training data (CCA, MCA and RDA), consistent with higher skill in isolating the known signal. Given that MCA maximizes the square of covariance leading to an amplitude bias, correcting this bias with SSVD yields stronger relationships that better resemble the known signals. Also, values of scaling associated with highest skill are shown to be proportionate to the signal-to-noise ratio. It follows that provided with some crude prior expectation of the signal-to-noise ratio, one may appropriately choose scaling values. Interestingly, very small but nonzero scaling values (nearing those of CCA but not CCA) are also found to produce strong results.

Following the synthetic experiments, SSVD is applied to examine tropical-extratropical teleconnections, and SSVD solutions are shown to produce robust relationships for ENSO and ENSO Modoki teleconnections that are stronger compared with those estimated by traditional methods after cross-validation. It is found that SSVD solutions using very small scaling values yield the strongest relationships. Following this, an additional application of SSVD to statistical downscaling is investigated with the goal of enhancing skill of MME seasonal prediction of precipitation over the Maritime Continent. Different SSVD solutions are shown to produce a range of skill, with some solutions yielding the highest skill in contrast to other methods, providing an improvement in prediction skill for least the inter-annual component of precipitation variability. Further application to statistical downscaling is proposed for future study, for which SSVD and related pattern-based methods show some promise.

1. INTRODUCTION

2. SCALED COVARIANCE

3. SCALED SVD ANALYSIS (SSVD)

3.1 SSVD as a constrained optimization problem

3.2 SSVD maximization and basis sets

3.3 A CCA-like variant of SSVD

4. SYNTHETIC EXPERIMENTS

5. APPLICATION TO TROPICAL-EXTRATROPICAL TELECONNECTIONS

5.1 Datasets & methodology

5.2 Results

6. PRELIMINARY RESULTS OF STATISTICAL DOWNSCALING

7. SUMMARY AND CONCLUSIONS