to (6.44), the proportion of Y realizations that are

p and q might change drastically when the sample

greater than X q is p. Thus the de¬nition states that

two random variables X and Y are (q, p)-recurrent size increases, but the main conclusions, that the

two distributions are very well separated and that

if there is a point between f X and f Y such that

the probability of misclassi¬cation is small, are not

proportion q of all X realizations lie to the left of

likely to change.21

X q and proportion p of all Y realizations lie to

the right of this point. If p and q are close to 1,

then the two random variables are almost perfectly6.9.5 Non-uniqueness of the Numbers p and q.

separated. On the other hand, if the distributionsThe point of separation X q in Figure 6.13 may

are symmetrical and p = q = 0.5, then the means be shifted; thus (q, p)-recurrence is equivalent

are equal. to (q , p )-recurrence for an in¬nite number of

pairs (q , p ). In particular, there is always one

6.9.3 Classi¬cation. Another way to understand number p so that (q, p)-recurrence is equivalent

the idea of (q, p)-recurrence is to think of a to ( p , p )-recurrence.

classi¬cation problem. Let us assume that we 21 Note, however,

have a pair of random variables X and Y that MVE outbreaks has that the relationshipdiscovery the the link

between SO and

changed since the of

are (q, p)-recurrent, and a realization z that is because precautionary measures are now taken to control

drawn from either X or Y. We want to determine outbreaks when the SO index is low.

6: The Statistical Test of a Hypothesis

124

A reasonable estimator of p-recurrence can be

obtained from (6.45) by replacing µ X , µY , and σ

with the corresponding estimators X, Y, and S p ,

where S 2 is the pooled sample variance (6.17).

p

Then

Y’X

p = FN

ˆ . (6.47)

Sp

6.9.7 Testing for (q, p)-recurrence. To test

that the response to experimental conditions is at

least (q, p)-recurrent, we assume that we have

n X realizations x1 , . . . , xn X of the control state

X, n Y realizations y1 , . . . , yn Y of the experimental

state Y, and that all realizations are mutually

statistically independent. The null hypothesis is

that X and Y are less than (q, p)-recurrent, that

is

Figure 6.14: Frequency distribution of annually

averaged (March to February) Darwin sea-level H0 : P Y > X q < p. (6.48)

pressure for seven years when Murray Valley

Two classes of tests are suggested in [404]:

encephalitis (MVE) was reported and for 63 years

one is a parametric test based on the assumption

when no cases of MVE were reported. The two

of normality and the other is a non-parametric

distributions are estimated to be (86%, 95%)-

permutation test. We present the parametric test in

recurrent.

the next subsection.

6.9.8 A Parametric Test. To construct a

If random variables X and Y have identical

parametric test we adopt a statistical model for

symmetrical distributions except for their means,

the random variables X and Y, namely that both

then (q, p)-recurrence is equivalent to ( p, q)-

random variables are normally distributed with

recurrence.

identical variances σ 2 . Using (6.46), we formulate

the null hypothesis H0 that the response is less than

6.9.6 p-recurrence. Recall that p-recurrence p-recurrent as

is synonymous with (0.5, p)-recurrence. Suppose µY ’ µ X

< Z p.

now that both random variables X and Y are H0 : (6.49)

σ

normally distributed with means µ X and µY and

If the null hypothesis is valid, the standard

a common standard deviation σ . If X and Y are

p-recurrent (with p ≥ 0.50 so that µ X < µY ), t-statistic (6.15) has a non-central t distribution

(see Pearson and Hartley [307]) with n X + n Y ’ 2

then

degrees of freedom and a non-centrality parameter

such that

p = P (Y > µ ) X

Zp

µ X ’ µY < . (6.50)

= 1 ’ FN

σ +

1 1

nX nY

µY ’ µ X

= FN , (6.45) Therefore, to test H0 we compute the usual t-

σ

statistic (6.16)

where FN is the distribution function of the ¯ ¯

Y’X

standard normal distribution N (0, 1). Thus the t = .

S p n X + nY

1 1

difference between X and Y is p-recurrent when

µY ’ µ X If 1 ’ p is the acceptable risk of erroneously

˜

= Z p, (6.46)

σ rejecting the null hypothesis, this t-value is

˜

compared with the p percentile, tn X +n Y ’2, ,˜ , of

p

’1

where Z p = FN ( p) (see Appendix D). the non-central t distribution with (n X + n Y ’ 2)

6.9: Univariate Recurrence Analysis 125

except that the non-centrality parameter in

(6.50) is replaced by

2Z p

= . (6.53)

+

1 1

nX nY

6.9.10 A Univariate Analysis: The Effect of

Cold Equatorial Paci¬c SSTs on the Zonally

Averaged Atmospheric Circulation. In June

1988, cold surface waters were observed in the

Eastern and Central Equatorial Paci¬c. This event

attracted interest in the scienti¬c community

because of its timing (northern summer) and

strength (these were the coldest June conditions

in the last 60 years). A numerical experiment

was performed to quantify the effect of such

Figure 6.15: The cross-section of monthly mean

anomalous lower boundary conditions on the

zonally averaged vertical ˜velocity™ [ω] in a paired

atmospheric circulation. Two 11-month perpetual

AGCM experiment on the effect of the anomalous

July simulations were performed: once with

SST conditions in June 1988. The contours lines

standard sea-surface temperatures and once with

show the 11-sample difference between the ˜June

the anomalous June 1988 SST distribution

1988 SST anomaly™ run and the ˜control™ run

superimposed (von Storch et al. [398]).

[398].

Monthly mean cross-sections of the zonally

Top: The points for which the null hypothesis of

averaged vertical ˜velocity™ [ω] obtained in the

equal means can be rejected with a standard t

two simulations were compared with univariate