recurrence analysis. The difference between the

shading) signi¬cance level.

mean [ω] cross-sections is shown in Figure 6.15.

Bottom: Points at which the univariate estimate

of (0.5, p)-recurrence (6.47) is less than 20% or Shading in the upper panel shows where the

difference of means is signi¬cantly different from

greater than 80% are shaded.

zero at the 5% (light) and 1% (dark) levels.

Clearly there is very strong evidence of change

in the mean Hadley circulation. On the other

degrees of freedom and non-centrality parameter hand, the lower panel in Figure 6.15 shows that

(6.50). These percentiles are given in [307] two [ω] distributions overlap substantially, even

and also in some statistical software libraries in the tropics. Regions are shaded where the

(e.g., IMSL [193]). For large sample sizes, the response is more than 80%-recurrent or less than

percentiles can be approximated by 20%-recurrent. There were no locations at which

the response to the anomalous SSTs was more

= + Zp. than 95%-recurrent or less than 5%-recurrent,

(6.51)

tn X +n Y ’2, ,˜ ˜

p

indicating that the anomalous SST does not excite

a response strong enough to eliminate the overlap

between the two density functions.

The physical message of the lower panel of

6.9.9 ( p, p)-recurrence. The multivariate gen- Figure 6.15 is that the inclusion of the anomalous

eralization of the concept of recurrence in Sec- tropical SST markedly modi¬es the Hadley cell

tion 6.10 requires ( p, p)-recurrence. Under the but that the atmospheric circulation poleward of,

conditions of [6.9.6], that is, both distributions are say, 20—¦ latitude is not affected by the anomalous

normal with the same variance, ( p, p)-recurrence forcing.

is equivalent to In this case the upper panel gives roughly

the same message; there is not much difference

µY ’ µ X between locations where there are signi¬cant

≥ 2Z p . (6.52)

σ differences (upper panel, Figure 6.15) and where

there is substantial recurrence. However, when

To test the null hypothesis that Y and X are less samples are larger, the estimated recurrence

than ( p, p)-recurrent, we proceed as in [6.9.8] generally gives a clearer indication of physically

6: The Statistical Test of a Hypothesis

126

signi¬cant responses than the local signi¬cance

test, since the rate of rejection in the latter is

sensitive to sample size.

6.10 Multivariate Recurrence

Analysis

6.10.1 Motivation. We described univariate

recurrence analysis as a classi¬cation problem in

[6.9.3]. Speci¬cally, if a realization z is drawn

randomly from X or Y, then the probability of

incorrectly determining the origin of z is 1 ’ p

when X and Y are ( p, p)-recurrent.

Figure 6.16 illustrates two bivariate normal

distributions X and Y with overlapping density

functions. We want to quantify this overlap in the

multivariate recurrence analysis, so we divide the

full two-dimensional plane into two disjoint sets

X and Y so that

P X∈ =P Y∈ =1’ p

Y X

(6.54)

P X∈ =P Y∈ = p.

X Y

The probability of a misclassi¬cation is then 1’ p.

The sets X and Y are easily found when X

and Y are multivariate normal [2.8.9] and have

the same covariance matrix Σ. The solution in

Figure 6.16: The density functions f X and f Y of

our bivariate example is sketched in Figure 6.16

two bivariate normal random variables X and Y

(bottom); X lies above the straight line and Y

below. In this example, p = 87.6%. In general, that differ only in their mean values.

Top: Three-dimensional representation of

when X and Y are of dimension m, X and Y are

max( f X , f Y ).

separated by an (m ’ 1)-dimensional hyper-plane.

Bottom: Contour lines of constant densities

We now sketch the basic ideas of multivariate

max( f X , f Y ) in the two-dimensional plane.

recurrence analysis. A more involved discussion

The straight line separates the full spaces into

of this approach can be found in Zwiers and

the two subsets X and Y so probability of

von Storch [452]. An application can be found in

misclassi¬cation is 1 ’ p = 12.4%.

Hense et al. [175].

From Zwiers and von Storch [452].

6.10.2 The Discrimination Function and the

Probability of Misclassi¬cation. The line (or The discriminating function is used to identify

more generally, hyper-plane) in Figure 6.16 the source of z when it is drawn randomly from

either X or Y. When W (z) ≥ 0, z is classi¬ed as

(bottom) that de¬nes the sets X and Y is given

by z = W ’1 (0) where W (·) is the discrimination being drawn from X and vice versa when W (z) is

function22 negative. The probability of correctly classifying Z

is

W (z) = zT Σ’1 (µ X ’ µY ) (6.55)

P Y∈ Y =P X∈ =p

X

1 T ’1

’ (µ X ’ µY ) Σ (µ X ’ µY ).

where p is given by

2

p = FN (D/2) ,

The sets and are then given by (6.57)

X Y

and D is the Mahalanobis distance [6.5.4],

= W ’1 ([0, +∞))

X

D2 = (µ X ’ µY )T Σ’1 (µ X ’ µY ).

= W ’1 ((’∞, 0)) . (6.58)

(6.56)

Y

D is a dimensionless measure of the distance

22 The discrimination function is used in multiple discrimi-

between the means of X and Y.

nant analysis (see Anderson [12], for example).

6.10: Multivariate Recurrence Analysis 127

m ’ 1 (n X ’ 3n Y )(n X + n Y ’ 2)

De¬nition of Multivariate ( p, p)-

6.10.3 ’ ’ D2 ,

S

4D S n X nY

Recurrence. Let X and Y be independent multi-

variate random vectors with identical covariance

where D S is the ˜shrunken™ Mahalanobis distance

matrices Σ X = ΣY = Σ and mean vectors

µ X and µY that are separated by Mahalanobis nX + ny ’ m ’ 3

D2 = Dˆ (6.60)

distance D (6.58). Then the difference between X n X + nY ’ 2

S

and Y is said to be ( p, p)-recurrent when p =

Dˆ = (Y ’ X)T Σ(X ’ Y), (6.61)

FN (D/2) (6.57).

In contrast with the univariate de¬nition, the and f N is the standard normal density function.

de¬nition above is restricted to multivariate normal

distributions with identical covariance matrices.

6.10.5 Testing for ( p, p)-recurrence. A para-

The concept is not easily extended to other mul-

metric test of ( p, p)-recurrence can be constructed

tivariate settings because, in general, derivation of

following the ideas of the parametric test of uni-

the surface that separates X from Y becomes

variate recurrence in [6.9.9]. The null hypothe-

intractible. For the same reason, (q, p)-recurrence