Figure 17.15: The response function of Black- Figure 17.17: Wallace et al.™s high-pass ¬lters

(Table 17.1, columns 7 and 8).

mon™s ¬lters (Table 17.1, columns 2 to 4).

17.5.7 Construction of Filters. Representation

(17.42) of the response function may be used to

choose ¬lter weights so that the resulting response

function c(ω) closely approximates a speci¬ed

˜

form c(ω). The weights for a speci¬ed ¬lter length

2K + 1 are obtained by minimizing

1

2 2

= c(ω) ’ c(ω) dω

˜

’1

2

1 K 2

2

= a0 + 2 ak cos(2πkω) ’ c(ω) dω.

˜

’1 k=1

2

Taking derivatives with respect to a0 and ak , and

Figure 17.16: Two ¬lters for isolating low- setting the derivatives to zero, we ¬nd

frequency and baroclinic variations for daily data 1

2

ak = ˜

(Table 17.1, columns 5 and 6). c(ω) cos(2π kω) dω

’1

2

for all k. Thus the ˜optimal™ ¬lter with 2K + 1

are also given in Table 17.1. The response weights is formed simply by truncating the Fourier

functions are shown in Figure 17.16. ˜

transform of c(ω). However, as we saw above,

the ˜optimal™ 2K + 1 weight ¬lter may have

A further pair of high-pass ¬lters, from Wallace

et al. [410], are listed in Table 17.1 and shown in undesirable properties, such as large side lobes, if

˜

Figure 17.17. When applied to daily observations c(ω) has discontinuous low-order derivatives. The

˜best™ 2K +1 weight ¬lter will be found by striving

the cut-offs are approximately 5 and 10 days.

˜

for a response function c(ω) that varies smoothly

with ω for all ω ∈ [’1/2, 1/2].

17.5.6 Examples. Several examples of applica- A strategy that results in good ¬lters is to

tions of ¬lters similar to those described above taper the weights of the optimal ¬lter that is

are discussed in Section 3.1. The day-to-day vari- obtained by truncating the Fourier transform of

˜

ability of DJF 500 hPa height is separated into c(ω) with a Hanning taper (see [12.3.8]). For

three time windows by means of Blackmon™s ¬lters example, consider a low-pass ¬lter with cut-off

frequency ω0 . The ˜optimal™ 2K + 1 weight ¬lter

(Figure 17.15) in [3.1.5], and the Northern Hemi-

has weights a0 = 2ω0 and ak = π|k| sin(2π |k|ω0 )

1

spheric distributions of the variances attributed to

for 0 < |k| ¤ K . The variance leakage problems

the three windows are shown in Figure 3.8. The

associated with this ¬lter are largely eliminated

skewness of the low-pass ¬ltered 500 hPa height

when these weights are tapered with factors h k =

and its relationship to the location of the storm-

2 (1 + cos(π|k|/(K + 1)) and then renormalized.

1

tracks is discussed in [3.1.8] (see Figure 3.11).

17: Speci¬c Statistical Concepts

390

The resulting low-pass ¬lter shuts off smoothly

0.0 0.2 0.4 0.6 0.8 1.0

with increasing frequency. The amplitude of the

5

variations at frequency ω0 are attentuated by 50%

10

20

Response

with the result that only 25% of the variance at this

40

frequency is passed by the ¬lter. The sharpness of

the cut-off is determined by the number of weights.

Figure 17.18 displays the response function for

¬lters with cut-off frequency ω0 = 0.1 and 11, 21,

41 and 81 weights (i.e., K = 5, 10, 20, and 40).

0.0 0.1 0.2 0.3 0.4 0.5

Frequency

Figure 17.18: Low-pass ¬lters with cut-off fre-

quency ω0 = 0.1 with 11, 21, 41 and 81 weights

(i.e., K = 5, 10, 20, and 40) constructed by

tapering the weights of the ideal low-pass ¬lter.

18 Forecast Quality Evaluation

18.0.1 Summary. Here we continue a discus- chapter and assume that P represents the true

sion that we began in [1.2.4], by extending our verifying observations.

treatment of some aspects of the art of forecast The method used to produce the forecast is

evaluation.1 We describe statistics that can be used not important in the context of this chapter.

to assess the skill of categorical and quantitative The forecast may have been produced using a

forecasts in Sections 18.1 and 18.2.2 The utility sophisticated dynamical model, but it may also

of the correlation skill score is discussed by il- have been based on a coin tossing procedure. The

lustrating that it can be interpreted as a summary information used to produce a forecast, called the

statistic that describes properties of the probability predictor, is also not relevant here.

distribution of future states conditional upon the A forecast must be precise in time and space.

That is, the time t for which the forecast F„ (t)

forecast. The Murphy“Epstein decomposition is

used to explain the relationships between com- is issued must be clearly stated and it must

monly used skill scores (Section 18.3). Some of correspond precisely with the time t of the

the common pitfalls in forecast evaluation prob- verifying analysis P(t). Thus, statements of the

lems are discussed in Section 18.4. form ˜there will be a thunderstorm at the end of

August™ do not qualify as a forecast.

18.0.2 The Ingredients of a Forecast. In this

18.0.3 Forecasts and Random Variables. In

section, we consider the problem of quantifying

this chapter, both the predictand P and the

the skill of a forecast such as that of monthly

forecast F are treated as random variables.

mean temperature at a certain location. We use the

An actual predictand, or actual observation, is

symbols F„ (t) to denote the forecast for the time t

denoted p, that is, as a realization of the random

with a lead time of „ (e.g., in units of months) and

variable P. Accordingly, a single forecast is

P(t) to denote the verifying observations, or the

denoted f.

predictand at time t. We generally omit the suf¬x

Skill parameters that measure the ensemble

(t) and the index „ in our notation unless they are

quality of forecasting system are parameters that

needed for clarity.

characterize some aspect of the distribution of

Note that in some applications there may be

the bivariate random variable (F, P). In practice,

substantial differences between P and the true

where a skill parameter is derived from a ¬nite

observations. These differences might arise from