null hypothesis, S ∼ B(m, 1 ’ p) if local test Correlations between the winter (DJF) mean SO

˜

decisions are made independently of one another index and corresponding winter mean height

[6.5.2]; unfortunately, this usually doesn™t happen. anomalies were estimated at m = 936 grid

j

Livezey and Chen [257] suggested several points. The local null hypothesis H0 that the true

solutions to this problem. One approach is to correlation at grid point j is zero was tested at the

reduce the number of degrees of freedom [6.8.2] (1 ’ p) = 5% level at each of the 936 grid points

˜

(similar to the modi¬cation of the t test when using a method that accounts for serial correlation.

the data are serially correlated [6.6.3]). Another The local null hypothesis was rejected at 11.4%

is to use a series of Monte Carlo experiments to of grid points”that is, d d/m = 0.114. This

T

simulate the statistical properties of the random is substantially larger than the 5% frequency that

would be expected if all local null hypotheses were

variables that enter the local decisions [6.8.3].

correct.

Figure 6.12a illustrates the rejection frequency

6.8.2 Reduced Number of Spatial Degrees • = dTd/m required to reject the global null

of Freedom. In many applications the local hypothesis at a global 5% signi¬cance level as

decisions are made on a regular grid so that each a function of the number of independent spatial

point has approximately the same number of points degrees of freedom m — . The rejection frequency •

in its immediate neighbourhood. The observations is given by

used to test a local hypothesis at a grid point

m—

are often strongly correlated with those used at ˜

B(m — , 1 ’ p)( j) ≥ (1 ’ p).

˜ ˜

nearby neighbours and roughly independent of min •

j=•·m —

those at distant grid points. Then it may be

possible to select a subset of grid points so that We see that a local rejection rate of • = 11.4%

the observations at these grid points are mutually supports rejection of the global null hypothesis in

¬elds that have m — = 52 or more spatial degrees of

independent.

Suppose there are m grid points in total and that freedom. However, seasonal mean 700 hPa height

the size of the subset is m — . Let D be the vector of is a very smooth ¬eld with very large spatial

6: The Statistical Test of a Hypothesis

122

upon the observed sequence of height ¬elds.

The process of simulating the SO index and

computing the test statistic S was repeated 200

times. The resulting distribution function is shown

in Figure 6.12b. Note that 5% of all randomly

generated S statistics are greater than 12.5%. Thus

we again ¬nd that the global null hypothesis can

not be rejected at the 5% level.

6.9 Univariate Recurrence Analysis

6.9.0 Motivation. The t test was introduced in

Section 6.6 to test the null hypothesis, H0 : µ X =

µY , that a pair of univariate random variables X

and Y have equal means. The power of the test

depends upon two factors. It increases when the

˜signal™ µY ’ µ X increases, and when the sample

sizes n X and n Y increase. This is illustrated in

Figure 6.3, where we displayed the signals δ =

(µ X ’ µY )/σ for which a test conducted at the

5% signi¬cance level has power 50% and 90%

given sample sizes n X = n Y = n. Note that the

probability of rejecting H0 is 90% when δ = 0.5

and n = 100, but that it is less than 50% when

n = 20.

More generally we ¬nd, for all signi¬cance

levels (1 ’ p) and all signals δ = 0, that the

˜

Figure 6.12: probability of rejecting H0 converges to 1 as n ’

a) Estimated percentage of rejected local null ∞. Thus, paradoxically, poor scientists are less

hypotheses required to reject the global null likely to detect physically insigni¬cant differences

hypothesis (that all local null hypotheses are valid) than rich scientists (see [6.2.5]).

at the 5% level. From Livezey and Chen [257]. One solution to this problem is to use scienti¬c

b) Livezey and Chen™s example [257]. Monte knowledge to identify the size of signal that is not

Carlo estimate (200 trials) of the rate • of physically signi¬cant and then to derive a test that

erroneous rejections of local null hypothesis when rejects H0 only when there is evidence of a larger

the global null hypothesis is true. The hatched area signal. This is the idea behind recurrence analysis.

marks the 10 largest random S statistics so that the We introduce the univariate concept [404] in this

critical value κp is 12.5%. The value to be tested,

ˆ section, and the multivariate generalization [452]

S = 11.4%, is marked by the lag-0 arrow. in Section 6.10.

Applications of the recurrence analysis include

[141, 175, 223, 404, 452].

covariance structures, so it is unlikely that this ¬eld

contains as many as 52 spatial degrees of freedom.

6.9.1 De¬nition. Two random variables X and

Hence there is insuf¬cient evidence to reject the

Y are said to be (q, p)-recurrent if

global null hypothesis.

Livezey and Chen [257] also describe an P Y > X q = p, (6.44)

attempt to use Monte Carlo methods to estimate

the distribution of S = DTD/m under the where X q is the qth quantile of the random

variable X.

global null hypothesis. The authors conducted

the Monte Carlo experiment by replacing the In many climate modelling applications X rep-

SO index time series with a random (˜white resents the control climate and Y represents a

noise™) time series. This ensured that all local climate disturbed by anomalous boundary condi-

correlations were zero. The authors did not tions or modi¬ed parameterizations of sub-grid

simulate the 700 hPa height ¬elds. Thus, the scale processes. The word recurrence refers to the

probability p of observing Y > X q . The strength

reference distribution they obtained is conditional

6.9: Univariate Recurrence Analysis 123

which population z was actually drawn from.

Furthermore we want to know the probability of

making an incorrect decision.

Area p

Area q

The decision algorithm is:

• ˜z is drawn from X™ if z < X q

• ˜z is drawn from Y™ if z ≥ X q

If z is really drawn from X then P z < X q = q

so that the probability of a correct decision is q in

this case. On the other hand, if z is really drawn

from Y, then by (6.44) P z > X q = p so that

the probability of a correct decision is p. The

Xq

probabilities of incorrect decisions are 1 ’ q and

1 ’ p, respectively.

Figure 6.13: De¬nition of (q, p)-recurrence: the

˜control™ represents the random variable X and 6.9.4 The Murray Valley Encephalitis Ex-

the ˜experimental™ the random variable Y. The ample. Before we discuss mathematical aspects

size of the area hatched to the left is q, so that of recurrence, we present a concrete example of a

P X < X q = q. The size of the area the hatched recurrence analysis.

to the right is p, and P Y > X q = p [404]. Between 1915 and 1984 there were seven

outbreaks of Murray Valley encephalitis (MVE)

in the Murray Valley in southeast Australia.

of the effect of the anomalous boundary conditions The prevalence of MVE virus depends on the

or modi¬ed parameterizations is measured against abundance of mosquitos, which in turn depends

the reference value q. on climate. Nicholls [292] studied the relationship

In many applications q = 50% so that the between the appearance of MVE and the state of

reference level is the mean of X. In that case we the Southern Oscillation (see [1.2.2]), and found

simply speak of p-recurrence. that annual mean sea-level pressure at Darwin was

unusually low in all seven MVE years.

6.9.2 Illustration. The idea of the (q, p)- The frequency histograms of annually averaged

Darwin pressure in MVE and non-MVE years are

recurrence is illustrated in Figure 6.13, where X q

plotted in Figure 6.14. The random variables X

represents a point on the right hand tail of f X .

(Darwin pressure conditional on the presence of

By de¬nition, the proportion of X realizations that

MVE), and Y (Darwin pressure conditional on

are less than X q is q. This point also represents

the absence of MVE) are highly recurrent, with

a point on the left hand tail of f Y and, according