consisting of daily analysis since approximately

1900, was available for determining the skill of the 14.4.1 Introduction. So far, we have identi¬ed

pairs of patterns by maximizing the correlation

PPP model.

The PPP analysis was performed with daily 11 This measure of skill is explained in detail in [18.2.9].

winter SLP maps for 1958“88. The dimensionality Roughly speaking, it is the mean spatial correlation between

of the problem was reduced by projecting the maps the forecast and the verifying ¬eld.

14: Canonical Correlation Analysis

328

are strongly linked through a regression model.

Patterns are selected by maximizing predictand

variance. This technique was developed in the late

1970s but apparently has not been introduced in

climate research literature.

Here we present the redundancy analysis as

suggested by Tyler [376]. Note that very little

experience has been collected with this technique

in the ¬eld of climate research. Therefore, the

I technique should be applied with great care, and

results should be appraised critically.

14.4.2 Redundancy Index. Let us consider a

pair of random vectors (X, Y) with dimensions m X

and m Y . Let us assume further that there is a linear

operator represented by a m X — k matrix Qk . How

much variance in Y can be accounted for by a

regression of QT X on Y?12 We assume, without

k

loss of generality, that the expected value of both

II X and Y is zero.

The regression model that relates QT X is given

k

Figure 14.8: Anomaly correlation coef¬cient of by

the PPP forecast (diamonds) and of persistence

(triangles).The vertical bars indicate ±σ bands, Y = R(Qk X) + ,

T

(14.29)

as estimated from all forecast prepared for

the winter days from 1900 until 1990. For where R is an m Y — k matrix of regression

better readability, the numbers for the two coef¬cients. The variance represented by (Qk X)

T

forecast schemes, persistence and PPP, are shifted is maximized when

horizontally. ’1

R = ΣY,QX ΣQX,QX , (14.30)

Top: For lags „ = 1, . . . , 5 days.

Bottom: For lag „ = 3 days. The anomaly where

correlation coef¬cients were classi¬ed according

ΣY,QX = Cov Y, QT X = ΣY X Qk

to the proportion of variance of the initial SLP

k

¬eld described by the PPP (bottom). Class 1

(14.31)

contains cases with proportions in the range

[0.0, 0.4], class 2 contains cases with proportions ΣQX,QX = Qk Σ X X Qk .

T

(14.32)

in (0.4, 0.5], and so on up to class 7, which

Tyler [376] called the proportion of variance rep-

contains cases with proportions in (0.9, 1].

resented by the regression (14.29) the redundancy

index and labelled it

between the corresponding pattern coef¬cients.

R 2 (Y : QT X) = (14.33)

We then demonstrated how regression techniques k

can be used to specify or forecast the value of

tr Cov(Y, Y) ’ Cov(Y ’ Y, Y ’ Y)

the pattern coef¬cients of one of the ¬elds from

tr Cov(Y, Y)

those of the other ¬eld. This regression problem is

generically non-symmetric because the objective is

to maximize the variance of the predictand that can where Y = R(QT X) is the estimated value of

k

be represented. Properties of the predictor patterns, Y. The motivation of this wording is that it is a

such as the amount of variance they represent, measure of how redundant the information in Y is

are irrelevant to the regression problem. Hence, if one already has the information provided by X.

there is a mismatch between CCA, which treats

12 The number of columns (patterns) in Q is smaller than

variables equally, and regression analysis, which k

the dimension of X in most practical situations, so that k < m X

focuses primarily on the predictand. m X . Thus, the operation X ’ QT X represents a

or even k k

The ˜redundancy analysis™ technique directly reduction of the phase space of X, as in all the other cases we

addresses this problem by identifying patterns that have discussed in this and the previous chapter.

14.4: Redundancy Analysis 329

provided that ˜column spaces™13 of Qk , Qk+1 , and

The numerator is the trace (sum of main

Qm X are nested and Qm X is invertible. If, for all

diagonal elements) of the matrix

k, Qk+1 is constructed by adding a column to Qk ,

ΣY Y ’ (ΣY Y + ΣY Y ’ 2ΣY Y )

ˆˆ ˆ then inequality (14.37) simply re¬‚ects the fact that

= ’ RΣQX,QX R + 2ΣY,QX R .

T T the regression on Y has k predictors in the case of

Xk , and the same k predictors plus one more in the

Using (14.31)“(14.33), and simplifying, we ¬nd case of X .

k+1

that For a given transformation Qk , again only the

subspace spanned by the columns of Qk matters.

R 2 (Y : QT X) = (14.34)

That is, for any invertible k — k matrix L, we ¬nd

k

’1 T

tr ΣY X Qk Qk Σ X X Qk Qk Σ X Y

T

. R 2 Y : LT (QT X) = R 2 (Y : QT X). (14.38)

k k

tr ΣY Y

Thus, the redundancy index for two variables

is a function of the subspace the variable X is

14.4.3 Invariance of the Redundancy Index projected upon, and the way in which Y is scaled.

Since R 2 does not depend on the speci¬c

to Linear Transformations. The redundancy

index has a number of interesting properties. coordinates of the variable X and Xk , we may

One of these is its invariance to orthonormal assume that the columns of Qk are chosen to be

orthogonal with respect to X; that is,

transformations of Y: if A is orthonormal, then

T

=0

q k ΣX X q j

R 2 (AY : QT X) = R 2 (Y : QT X). (14.39)

(14.35)

k k

for any k = j. Then

The signi¬cance of this property comes from

the fact that we may identify any orthonormal k

j T X),

R (Y : QT X) = R 2 (Y : q

2

transformation with a linear transformation that (14.40)

k

conserves variance. Relationship (14.35) does j=1

not hold for general non-singular matrices, in

which may be seen as a special version of (14.37).

particular not for transformations that change the

Note that (14.39) is ful¬lled if the vectors q j are

variance since the proportion of captured variance

the EOFs of X.

changes when the variance of Y is changed.

On the other hand, any square non-singular

matrix Qm X used to transform the specifying 14.4.4 Redundancy Analysis. The theory

behind redundancy analysis, as put forward by

variable X has also no effect on the redun-

dancy index. In that case, (Qm X )’1 exists and Tyler [376], con¬rms the existence of a non-

singular transformation B = (b1 |b2 | · · · |bm X ) so

(QT X Σ X X Qm X )’1 = Q’1 Σ’1 (QT X )’1 in the