Zoltan
Toth, Olivier Talagrand, Guillem Candille and Yuejian Zhu
Draft
(July
11,
2002)
7.1
Introduction
The
previous chapters discussed verification procedures with respect to environmental
predictions that are expressed in the form of a single value (out of a
continuum) or a category. This chapter is devoted to the verification of
probabilistic forecasts, typically issued for an interval or a category.
Probabilistic forecasts differ from the previously discussed form of predictions
in that, depending on the expected likelihood of forecast events, they
assign a probability value between 0 and 1 to possible future states.
It
is well known that all environmental forecasts are associated with uncertainty
and that the amount of uncertainty can be situation dependent. Through
the use of probabilities the level of uncertainty associated with a given
forecast can be properly conveyed. Probabilistic forecasts can be generated
through different methods. By considering a wide range of information,
forecasters can subjectively prepare probabilistic forecasts. Alternatively,
statistical techniques can be used either on their own, based on observational
data (see, e. g., Mason and Mimmack, 2002; Chatfield, 2000), or in combination
with a single dynamical model forecast and its past verification statistics
(Atger, 2001).
Probabilistic
forecasts can also be based on a set of deterministic forecasts valid at
the same time. Assuming the forecasts are independent realizations of the
same underlying random process, the best estimate of the forecast probability
of an event is equal to the fraction of the forecasts predicting the event
among all forecasts considered. Ensemble forecasting, as this technique
is called (see, e. g., Leith 1974; Ehrendorfer 1997), can thus produce
probabilistic forecasts based on a set of deterministic forecasts, without
necessarily relying on past verification statistics. In certain fields
of environmental sciences, such as meteorology and hydrology, the ensemble
forecasting technique is widely used. Therefore methods used to directly
evaluate a set of ensemble forecasts, before they are interpreted in probabilistic
terms, will also be discussed in this chapter.
In
the next section we will discuss the two main attributes of probabilistic
forecasts, usually referred to as "reliability" and "resolution". Sections
7.3 and 7.4 will introduce a set of basic verification statistics that
can be used to measure the performance of probabilistic forecasts for categorical
and continuous variables with respect to these attributes. Section 7.5
is devoted to measures of ensemble performance, while section 7.6 discusses
some limitations to probabilistic and ensemble verification. Some concluding
remarks are found in section 7.7.
7.2.
The Main Attributes of Probabilistic Forecasts: Reliability and Resolution.
How
can one objectively evaluate the quality of probabilistic forecasts? Let
us consider the following prediction: «There is a 40% probability
that it will rain tomorrow». Assuming that the event 'rain' is
defined unambiguously, it is clear that neither its occurrence nor its
nonoccurrence can be legitimately used to validate, or invalidate the
prediction. Obviously, a probabilistic forecast cannot be validated on
the basis of a single realization. This is in contrast with deterministic
forecasts (?it will rain? or ?it will not rain?), which can
be validated, or invalidated by one subsequent observation.
The
only way to validate a forecast expressed in terms of probabilities is
statistical. In the case above, one must wait until the 40 % probability
forecast has been made a number of times, and then check the proportion
of occurrences when rain was observed. If that proportion is equal or close
to 40%, one can legitimately claim the forecast to be statistically correct.
If, on the contrary, the proportion is significantly different from 40%,
the forecast is statistically inconsistent.
One
condition for the validity of probabilistic forecasts for the occurrence
of an event is therefore statistical consistency between a priori
predicted probabilities and a posteriori observed frequencies of
the occurrence of the event under consideration. Consistency of this kind
is required, for instance, for users who want to take a decision on the
basis of an objective quantitative risk assessment (see chapter 8). Following
Murphy (1973), this
property of statistical consistency is called reliability. Reliability
is the property that a probabilistic forecast is statistically valid.
Reliability,
however, is not sufficient for a probabilistic forecast system to be useful.
Consider the extreme situation where one would predict, as a form of probabilistic
forecast for rain, the climatological frequency of occurrence of rain.
That forecast system would be reliable in the sense that has just been
defined, since the observed frequency of rain would be equal to the (unique)
predicted probability of occurrence. But that system would not provide
any forecast information beyond climatology. As a second condition, one
requires that a forecast system reliably distinguish between cases with
lower and higher than climatological probability values. After Murphy (1973),
the ability of a forecast system to reliably vary forecast probability
values is called resolution. Reliability and resolution together
determine the usefulness of a probabilistic forecast system. There seems
to be no desirable property of probabilistic forecast systems other that
these two characteristics.
What
has just been described for probabilistic prediction of occurrence of events
easily extends to all forms of probabilistic prediction. Let us consider
the quantity x (for instance, temperature at a given time and location),
and a corresponding forecast probability density function p(x),
represented by the full curve in Figure
7.1. For a particular case, the subsequent verifying observation value
is shown by x_{o}
(see Fig. 7.1). If, as in our example, x_{o}
falls within the range of the forecast distribution, the observation does
not validate or invalidate the forecast.The
difficulty here is that, contrary to what happens with a singlevalue forecast
(see chapter 5), it is not possible to define, in a trivial way, a 'distance'
between the prediction (here the probability distribution p(x))
and the observation. Again, validation is possible only in a statistical
sense. One must wait until the distribution p(x) is predicted a
number of times, and then compare p(x) to the distribution p_{o}(x)
of the corresponding verifying observations. If p_{o}(x)
is identical with p(x) (or at least close to p(x) in some
sense, as the distribution p_{1}(x)
shown by the dashdotted curve in Figure 7.1), then the prediction p(x)
can legitimately be said to be statistically consistent, or correct. If,
on the contrary, p_{o}(x)
is distinctly different from p(x) (as the distribution p_{2}(x)
shown by the dashed curve in the figure), then the prediction p(x)
is statistically inconsistent.
This
example calls for a more general definition of reliability. A system that
predicts probability distributions is reliable if, for any forecast
distribution p, the distribution of the corresponding verifying
observations, taken when p was predicted, is identical to p.
This definition can also be extended to multidimensional and any other
type of probabilistic forecasts.
As
noted earlier, reliability, albeit necessary, is not sufficient for the
practical utility of a probabilistic forecast system. Systematic prediction
of the climatological distribution of a meteorological variable is reliable
yet it provides no added forecast value. Probabilistic forecasts should
be able to reliably distinguish among sets of cases (through the use of
a distinctly different forecast probability distribution, p^{i},
for each set) for which the distribution of corresponding observations
(p^{i}_{o}) are distinctly different (and therefore
also different from the climatological distribution). Such a system can
?resolve? the forecast problem in a probabilistic sense, and is called
to have resolution. The more variations in probability values a forecast
system can use in a reliable fashion, the more resolution it has. Maximum
resolution is obtained if the forecast distributions are reliable and concentrated
on one point, as Dirac functions. Note that a probabilistic forecast system
generating perfectly reliable probabilistic forecasts at maximum resolution
is actually a perfect categorical forecast system (see chapters 4 and 5).
A probabilistic approach to forecasting is warranted when maximum resolution,
due to the presence of uncertainties, is not attainable.
The
two main attributes of probabilistic forecasts, reliability and resolution,
are a function of both the forecasts and verifying observations. Sharpness
is another characteristic of probabilistic forecast systems. Resolution
was introduced above as reliable deviations of forecast probabilities
from the climatological probabilities. Unlike resolution, sharpness is
a function of the forecast (and not the observed) distributions only and
it simply measures how different the forecast probability values are from
climatological probabilities, irrespective whether those probabilities
are consistent with the corresponding observations. It follows that one
can easily make a probabilistic forecast sharper by bringing its shape
closer to a Dirac function. Such an increase in the probability values
for the mode of the distribution, however, generally will not enhance resolution.
Resolution can be improved upon only through a clearer separation of situations
where the event considered is more or less likely to occur as compared
to the climatological expectation.
Resolution
thus cannot be improved via a simple change of probability values. Such
improvements can be achieved only through the use of additional genuine
knowledge about the behavior of the system that is being predicted, by
changing the way groups of forecast cases are identified. This suggests
that resolution, i. e., the ability to reliably identify cases with higher
and lower than climatological expectancy, is in fact the intrinsic value
of forecast systems. This becomes even clearer when the effect of statistical
postprocessing is considered on the other, complimentary attribute of
probabilistic forecasts. Contrary to the case of resolution, reliability
can be improved by simple statistical postprocessing that modifies
the forecast probability values. Let us assume, e. g., that the forecast
distribution p(x) is associated with a distinctly different distribution
of observations p_{2}(x)
(dashed curve in Fig. 7.1). The next time the system predicts p(x),
one can use p_{2}(x)
as the calibrated forecast instead. This a posteriori calibration
will make a forecast system, that was not so in the first place, reliable.
For statistically stationary forecast and observed systems, perfect reliability
can always be achieved, at least in principle, by such an a posteriori
calibration. This, again, suggests that the intrinsic value of forecast
systems lies not in their reliability (that can be made perfect through
statistical adjustments) but in their resolution.
In
summary, a useful forecast system must be able to a priori separate situations
into groups with as different outcomes as possible, represented by a distinct
distribution of observations associated with each forecast group (resolution).
If such a forecast system exists, the different forecast outcomes (groups
of cases), even if originally designated differently, can be ?renamed?
and marked by the observed distributions that are associated with them
(reliability), based on a simple statistical verification and postprocessing
procedure.
In
the following section a number of scores that are used for the evaluation
of binary, multicategorical, and continuous variable probabilistic forecasts
will be introduced. These scores will be systematically analyzed as to
what they exactly measure in terms of reliability and resolution, and their
significance will be illustrated using recent meteorological applications.
In our analysis averages taken over all n available realizations
of a probabilistic forecast system, or over a subset of cases satisfying
a condition C will be denoted by E(.) and E_{c}(.)
respectively.
7.3.
Probabilistic Forecasts for Occurrence of Events.
In
the following three subsections we will consider verification tools for
the evaluation of probabilistic forecasts for the occurrence of a particular
event E
such as ?the temperature at a given location x at forecast lead
time t will be less than 0 C?, or ?the total amount of precipitation
over a given area and a given period of time will be more than 50 mm?.
We denote by f(q) the observed frequency of occurrence of E
in the circumstances when E is
predicted to occur with probability q (0 #
q#
1). The condition for reliability, as defined in the previous section,
is that f(q) = q for all q.
7.3.1.
The reliability curve.
As
an example let us consider probabilistic forecasts based on the National
Centers for Environmental Prediction (NCEP) Ensemble Forecast System (Toth
and Kalnay, 1997) for the event of the 850hPa temperature anomaly from
the 1999 winter sample climatological mean value
(T_{c}) being t
= 4C or less. Diagnostics are accumulated over all gridpoints located
between longitudes 90W and 45E, and between latitudes 30N and 70N, and
over 65 forecasts issued between 1 December 1998 and 28 February 1999,
for a total of n=16,380 realizations. Forecast probability values
are defined as i/m, where i (i = 0, ?, m)
is the number of members that predicted the event out of an m=16
member ensemble. The forecast probabilities are thus discretized to m+1
= 17 predefined values. For deciding if the event occurred or not, the
operational analysis of NCEP will be used as a proxy for truth (?verifying
observation?).
The
solid line in Fig. 7.2 shows a
reliability curve, depicting variations of f(q) as a function of q, for
the forecast system and event described above. Although rather close to
the diagonal f(q) = q, the reliability curve shows
some deviations from it. In particular, the slope of the reliability curve
in Fig. 7.2 is below that of the diagonal. Note that deviations from the
diagonal are not necessarily indicative of poor reliability. When statistics,
like in our example, are based on a finite sample the reliability curve
for even a perfectly reliable forecast system would exhibit fluctuations
around the diagonal. The effect of sampling variability can be quantified
by verifying the same ensemble again except this second time against one
of its randomly chosen members. In this setup, by definition, the forecast
system is perfectly reliable and any deviations from the diagonal are due
to sampling fluctuations caused by the finite volume of data. When compared
to the diagonal, the difference between the perfect (dashdotted line in
Fig. 7.2) and operational ensemble curve (solid line) reflect the actual
lack of reliability in the forecast system, irrespective of the size of
the verification sample.
The
bar graph in the lower right corner of Figure 7.2 is known as a sharpness
graph, and shows the frequency
p(q) with which each probability
q is predicted. The 0 and 1 probability values are used in 90% of
the realizations of the process. This means that, at the 2day range considered
here, the dispersion of the predicted ensembles is relatively small so
that in most cases all forecast temperature values in the ensemble fall
on the same side of the threshold T_{c }+ t
that is used to define the event. In a quantitative evaluation of the performance
of the forecast, the two extreme points p =0 and p =1 must
therefore be given proportionately more weight.
7.3.2.
The Brier Score.
Brier
(1950) proposed the following measure for the quantitative evaluation of
probabilistic forecasts for the occurrence of a binary event:(7.1)
where
n is the number of realizations of the forecast process over which
the validation is performed. For each realization j, q_{j}
is the forecast probability of the occurrence of the event, and o_{j}
is a binary observation variable that is equal to 1 or 0 depending on whether
the event occurred or not.
The
Brier score for a deterministic system that can perfectly forecast the
occurrence or nonoccurrence of events with q=1 or 0 respectively assumes
a minimum value of 0. On the contrary, the Brier score for a systematically
erroneous deterministic system that predicts probability 0 when the event
does, and 1 when it does not occur would assume a maximum value of 1.
For
comparing the Brier score to that of a competing reference forecast system
(B_{ref}) it is convenient to define the Brier Skill Score
(BSS):
BSS=1B/B_{ref} (7.2)
Contrary
to the Brier score (7.1), the Brier Skill Score is positively oriented
(i.e., higher values indicate better forecast performance). BSS
is equal to 1 for a perfect deterministic system, and 0 (negative) for
a system that performs like (poorer than) the reference system. Most often
climatology is used as a reference system. For a climatological
forecast system, i.e. a system that always predicts the climatological
frequency of occurrence p_{c}
of an event, o assumes the value 1 with frequency p_{c},
and the value 0 with frequency 1 p_{c}.
The Brier score is then:B_{c}=
p_{c}(1
p_{c}).
Such a system, as discussed before, has perfect reliability, but no resolution.
In the rest of this chapter, the Brier Skill Score will be defined with
the climatological forecast system as a reference (BSS_{c}=1B/B_{c}).
For
facilitating the forthcoming discussion, continuous variables are introduced
as:
(7.3)
where
p(q) is the frequency with which q is predicted. In
realistic forecast situations the number of distinct forecast probability
values is finite and the integral in (7.3) reduces to a finite sum. Since
when q is predicted the event occurs with frequency f(q),
the climatological frequency of occurrence of the event is
(7.4)
More
generally, integration with respect to p(q) is identical
with averaging over all realizations of the forecast process for any quantity
u:
(7.5)
The
statistical performance of a system for probabilistic prediction of occurrence
of an event is entirely determined by the corresponding functions p(q)
and f(q). Different prediction systems will correspond with
different functions of p(q) and f(q), while
the climatological frequency of occurrence p_{c},
that is independent of the prediction system, will in all cases be given
by Eq. (7.4).
From
now on, we will systematically express scores as explicit functions of
p(q) and f(q). On partitioning the average
in Eq. (7.1) according to the forecast probability q, the Brier
score B reads
(7.6)
For
given q, o assumes a value of 1 with frequency f(q),
and 0 with frequency 1 f(q). Therefore
E[(qo)^{2}½q]
= (q1)^{2}
f(q)
+ q^{2 }[1f(q)]
=[qf(q)]^{2}
+ f(q) [1f(q)](7.7)
from
which we obtain, upon integration with respect to
p(q)
dq,
and by using Eq. (7.4)
(7.8)
The
Brier score is thus decomposed into three terms, each with its own significance
(Murphy 1973). The first term on the right hand side of Eq. (7.8), that
is a weighted measure of the departure of the reliability curve from the
diagonal (see Fig. 7.2), is a measure of reliability. We recall that for
an ideal forecast system where events are observed with the same frequency
as they are forecast (f(q)= q) this term is zero. The second term (with
a negative sign) is a measure of how different the observed frequencies
corresponding to the different forecast probability values are from the
climatological frequency of the event. The larger these differences are,
the better the forecast system can a priori identify situations that lead
to the occurrence or nonoccurrence of the event in question in the future.
This term is a measure of resolution as defined in section 7.2. It is independent
of the actual forecast probability values (and thus also independent of
reliability), and is only a function of how the different forecast events
are classified (or ?resolved?) by a forecast system. The third term on
the right hand side of Eq. 7.8 is independent of the prediction system
and is a function of the climatological uncertainty only. The difficulty
(or lack of it) of predicting events with close to 0.5 (0 or 1) climatological
probability is recognized by a large (small) uncertainty term in Eq. 7.8.
The
scores(7.9a)
and
(7.9b)
will
be used to measure reliability and resolution respectively. Both scores
are negatively oriented, and are equal to 0 for a perfect deterministic
system. Like the Brier Skill Score BSS (Eq. 7.2), they are comparable
for events with different climatological frequencies of occurrence. They
are related to the Brier Skill Score (defined using the climatological
forecast system as a reference, BSS_{c}) by the equality
BSS_{c}
= 1  (B_{Rel }+
B_{Res})(7.10)
The
Brier score B for the operational NCEP system represented by the
solid curve in Figure 7.2 is equal to 0.066. The climatological frequency
of occurrence p_{c}
for the event under consideration is equal to 0.714, which gives a value
of 0.677 for the Brier Skill Score BSS_{c}. This corresponds
to B_{Rel }=
0.027 and B_{Res}
= 0.296.
These values are rather typical of the values produced by present short
and medium range Ensemble Prediction Systems. The fact that the reliability
term is significantly smaller (typically one order of magnitude less) than
the resolution term is generally observed. This fact, however, has no particular
meaning since, as pointed out earlier, reliability and resolution are two
different and independent aspects of forecast quality.
Figure
7.3 shows the Brier Skill Score defined using the climatological forecast
system BSS_{c} (full curve) and its two components B_{Rel }(shortdashed
curve) and B_{Res}
(dashed curve), as a function of forecast lead time, for the European Centre
for Mediumrange Weather Forecasts (ECMWF) Ensemble Prediction System (Molteni
et al. 1996). The event considered here is that the deviation of the 850hPa
temperature from its 1999 winter mean (sample climatology T_{c})
is less than t
= 2C. Scores were computed over the same geographical area and time period
as those used to construct Figure 7.2. Since no data were missing, the
total number of cases considered is now n = 22,680. Forecast probabilities
are defined, as in the case of Figure 7.2, as i/m (i
= 0, ?, m=50), and the verifying 'observation' is the ECMWF operational
analysis.
The
score BSS_{c}numerically
decreases (meaning the quality of the system degrades) with increasing
forecast range. The decrease is entirely due to the resolution componentB_{Res},
whereas the reliability component B_{Rel }(which,
as before, is significantly smaller than B_{Res})
shows no significant variation. The degradation of resolution corresponds
to the fact that, as the leadtime increases, the forecast ensemble has
a broader dispersion, and becomes more similar to the climatological distribution.
All these features are typical of current Ensemble Prediction Systems.
Finally
we note again that if both the forecast and observed systems are stationary
in time and there is a sufficiently long record of their behavior it is
possible to make an imperfect forecast system completely reliable (see
also section 7.2). If the frequency of occurrence of an event (f(q)) is
different from the forecast probability, the latter can be calibrated by
a posteriori changing it to f(q). This, if done on all values of q,
amounts to moving all points of the reliability curve horizontally to the
diagonal (Figure 7.2). As a result of this calibration, the first term
on the right hand side of Eq. (7.8) becomes 0, while the second term, that
measures the variations of the observed frequencies f(q)
around the climatological frequency as the forecast probability q changes,
is not modified. The observed frequencies f(q) are thus the
probabilities that can be effectively used by a forecast system, following
an a posteriori calibration. As mentioned earlier, resolution cannot
be improved through such a simple statistical correction of the forecast
probability values. This confirms that it is the resolution part of the
Brier score that measures a forecast system?s genuine ability to distinguish
among situations that lead to the future occurrence or nonoccurrence of
an event.
7.3.3.
Relative Operating Characteristics
Another
commonly used score for evaluating probabilistic forecasts of occurrence
of binary events is called the Relative Operating Characteristics
(or Receiver Operating Characteristics, Bamber, 1975), usually abbreviated
as ROC.
A
Relative Operating Characteristics graph is based on a set of contingency
tables used to evaluate deterministic forecasts for the occurrence
of a binary event. A contingency table is built by separating the realizations
of the system into four classes depending on whether the event under consideration
is predicted to occur or not, and on whether it actually occurs or not
(see also Chapter 3):
Observed









Forecast
where
A, B, C, and D are the number of realizations
in each of the four classes. In particular, the ratio
(7.11a)
is
the proportion of 'hits', i. e., of correctly predicted occurrences
of the event. The ratio
(7.11b)
is
the proportion of 'false alarms', i. e., of situations in which
the event is incorrectly predicted to occur.
The
Relative Operating Characteristics curve for a probabilistic prediction
system for the occurrence of an event is then defined as follows. For each
threshold value s, 0 #
s#
1, the probabilistic prediction is transformed into a deterministic prediction
by issuing a ?yes? forecast for the event if p is greater
than s, and a ?no? forecast otherwise. An mmember
ensemble, for example, can be fully described by the use of m thresholds
(corresponding to the distinct forecast events of at least 1, 2, 3, ...,
or m members predicting an event). The hit and false alarm rates
(Eq. 7.11) for the different values of s define the ROC curve.
When
the event is always predicted (s = 0, or strictly speaking,
s < 0, C = D = 0) then H = F = 1. At the
other extreme, when the event is never predicted (s = 1,
A = B = 0) H = F = 0. For any forecast system both
H(s) and F(s) decrease from 1 to 0 when s
increases from 0 to 1. The ROC curve for a climatological system that always
predicts the climatological frequency of occurrence p_{c}
is defined by the two points H(s) = F(s) = 1
for s < p_{c},
and H(s) = F(s) = 0 for s$
p_{c}.
In case of a deterministic forecast system (m=1) there is only one
decision criterion and the ROC curve is defined by the corresponding point,
along with the points at the end of the diagonal (H = F = 1, H
= F = 0). For a perfect deterministic system (B = C = 0) H(s)
= 1 and
F(s) = 0. A probabilistic forecast system with good
reliability and high resolution is similar to a perfect deterministic forecast
in that it will use probabilities that are all close to either 0 or 1.
For such a system all (F(s),
H(s)) points will
therefore be close to the optimal (0,1) point for a large range of values
of s. It follows that in general the proximity of the ROC curve
to the point (0, 1) is an indicator of the value of a forecast system.
Therefore the area below the curve in the (F, H)plane is
an overall quantitative measure of forecast value.
A
more detailed analysis of the significance of ROC results can be
given by first realizing that the relative frequency of hits in Table 7.1
(A) for a given probability threshold scan
be written, using earlier introduced notation, as:
Note
also that the sum A + C is equal to the climatological frequency
of occurrence p_{c}.
Therefore the hit rate can be written as:
(7.12a)
A
similar argument shows that the false alarm rate can be written as
(7.12b)
The
integral in (7.12a) is the average of f(q) for those circumstances
when q > s. If f(q) is a strictly increasing
function of q, that last inequality is equivalent to f(q)
> f(s), and the comparison with the threshold can be done
on the a posteriori calibrated probabilities f as well as
on the directly predicted probabilities q. The same argument applies
to the integral in (7.12b), which means that the ROC curve is invariant
to the a posteriori
calibration through which the points in the reliability curve are moved
horizontally to the diagonal. Thus, if the calibrated forecast probability
values f(q) are strictly increasing with the original forecast
probabilities q, the ROC curve, just like the resolution
component of the Brier score, depends only on the a posteriori calibrated
probabilities f (and their frequency distribution). The ROC
curve is therefore an indicator of the resolution of the prediction system
and is independent of reliability. The resolution component of the Brier
score and the ROC curve therefore provide very similar qualitative
information.
As
an example, ROC curves are displayed in Fig.
7.4 for the same event and forecast system as that studied in Fig.
7.3. Note that, as expected, the ROCarea defined by the curves
monotonically decreases as a function of increasing forecast lead time
(with values of 0.977, 0.952, 0.915 and 0.874 for 2, 4, 6 and 8 days lead
time respectively), just as the Brier score does in Fig. 7.3. While there
is a qualitative agreement in that both theROC
and Brier scores indicate a loss of predictability with increasing lead
time, the corresponding values for the two scores are quantitatively different.
Moreover, there is no onetoone relationship between the two measures.
It is not clear at present which measure of resolution, if either, is preferable
for any particular application. We note that the potential economic value
associated with the use of a forecast system is measured through the same
contingency table (Table 7.1) as that used to define ROC (see chapter
8).
7.3.4.
Information
content
It
has been argued that probabilistic forecasts made by and for stationary
processes can, at least theoretically, be made fully reliable through postprocessing.
It was also pointed out that the real value of forecasts lies in their
resolution. In this subsection, following Toth et al. (1998) we introduce
information content (I) as another statistic to measure the resolution
of reliable probabilistic forecasts. As an example, let us consider a set
of 10 climatologically equally likely intervals as predictands. In this
case it is convenient to define information content (I) in the forecast
based on log_{10} in the following way:
(7.13)
where
p_{i}is the forecast
probability of the i'th category. We require that and
0#
p_{i}#1 for
all p_{i}. If the forecasts are perfectly reliable, I
can be considered as another measure of resolution. Under these conditions
I varies between 0 (for a climatological forecast distribution)
and 1 (for a deterministic forecast distribution). I can be arithmetically
averaged over a set of forecast cases. Fig.
7.5 shows the information content of a 10member 0000 UTC subset of
the NCEP ensemble for 500 hPa height values as a function of leadtime.
For further examples on how the concepts of information theory can be used
in probabilistic forecast verification, the readers are referred to Stevenson
and DoblasReyes (2000), and Roulston and Smith (2002).
7.3.5
Multipleoutcome
events
The
Brier score (7.1) was defined above for the evaluation of probabilistic
forecasts for the occurrence or nonoccurrence of a single binary event.
While in many studies the Brier score is used as in Eq. 7.1, in his original
paper Brier (1950) gave a more general definition, considering multipleoutcome
events. Let us consider an event with K complete, mutually exclusive
(and not necessarily ordered) outcomes E_{k }(k
= 1, ?, K), of which one, and only one, is always necessarily
observed (e. g., 4 categories such as no precipitation, liquid, frozen,
or mixed precipitation). A probabilistic prediction for this set of events
then consists of a sequence of probabilities q = (q_{k}),
k = 1, ?, K, with S_{k }q_{k}
= 1. The score defined by Brier is
B_{K}
= (1/K) E[ S_{k=}1,K
(q_{k}o_{k})^{2 }]
= (1/K)
E[½½q
 o
½½^{2 }](7.14)
where
o_{k}
= 1 or 0 depending on whether the observed outcome is E_{k}
or not. On the right hand side of Eq. 7.14 ½½.½½ denotes
the Euclidean norm, ando
is the sequence (o_{k}).
B_{K}
is the arithmetic average of the Brier scores (7.1) defined for the various
outcomes E_{k},
each considered as binary events. A skill score, similar to (7.2) can be
defined as
BSS_{K }=
1  B_{K}
/ B_{refK}
The
reference Brier score (B_{refK}), for example, can be computed
for the climate forecast system that systematically predicts p_{ck}
for each k: B_{cK}
= (1/K) S_{k}
p_{ck}(1
p_{ck}),
where p_{ck}
is the climatological frequency of the occurrence E_{k}.
Examples for the use of the multievent Brier score can be found in Zhu
et al. (1996) and Toth et al. (1998).
A
reliabilityresolution decomposition of the score B_{K}
can be defined by averaging the decompositions of the elementary scores
for the occurrences E_{k}.
For multievent forecasts, a more discriminatory decomposition, built on
the entire sequence q
of predicted probabilities, seems preferable. Denoting dp(q)
the frequency with which the sequence q is predicted by the
system, and defining the sequence f(q) = [f_{k}(q)]
of the conditional frequencies of occurrence of the E_{k}'s
given that q has been predicted, a generalization of the
derivation leading to Eq. (7.8) shows that
(7.15)
where
p_{c}
is the sequence (p_{ck}).
Similarly to Eq. (7.8),Eq. (7.15)
provides a decomposition of B_{K}
into reliability, resolution, and uncertainty terms.
7.4.
Probabilistic Forecasts for a Numerical Variable.
While
the previous section presented basic verification tools for evaluating
single or multiple category binary events, here we discuss how probabilistic
forecasts for a numerical variable x, such as temperature given
at a point at a particular time, can be evaluated. Such a probabilistic
forecast consists of a full probability distribution, instead of one, or
a finite number of probability values. In any practical application, the
probability distribution will of course be still defined by a finite number
of numerical values. Note that the discretization here is needed only for
carrying out the numerical calculations, and is not imposed as a constraint
arising due to the format of the forecasts.
7.4.1
The Discrete Ranked Probability Score
Let
us define J thresholds x_{j}
(j =1, ?, J), x_{1}
< x_{2}
< ?< x_{J}, for
the variable x, and the associated J events H_{j}
= (x ? x_{j}),
(j = , ?, J). A probability distribution for x defines
a sequence of probabilities p_{1}
? p_{2}
? ?? p_{J }for
those events. The quantity o_{j}
is defined for each j as being equal to 1 or 0 depending on whether
the event H_{j}
is observed to occur or not. The discrete Ranked Probability Score (RPS)
is then defined as
(7.16)
where
B(x_{j})
is the Brier score (7.1) relative to the event H_{j}.
The
Ranked Probability Score is similar to the multievent Brier score (7.13),
but, as its name implies, it takes into account the ordered nature of the
variable x. Here the events H_{j}
are not mutually exclusive, and H_{j}
implies H_{j' }if
j < j'. Consequently, if x is predicted to fall
in an interval [x_{j},
x_{j}_{+1}]
with probability 1, and is observed to fall into another interval [x_{j}_{'},
x_{j}_{'+1}],
the RPS increases with the increasing absolute difference ½jj'½.
7.4.2
The Continuous Ranked Probability Score
An
extension of the RPS can be definedby
considering an integral of the Brier scores over all thresholds x, instead
of an average of Brier scores over a finite number of thresholds as in
(7.16). We denote by F(x) the forecast probability distribution
function, and x_{O}the
verifying observation. By choosing a measure m(x)
for performing the integral, the Continuous Ranked Probability Score (CRPS)
reads as:(7.17)
where
H is the Heaviside function (H(x)
= 0 if x <
0, H(x)
=1 if xC0).
Boththediscrete
and continuous Ranked Probability Scores, just
like the multievent Brier score (7.14), can be expressed as skill scores
(see Eq. 7.2), and are amenable to a reliabilityresolution decompositions.
For additional related information the reader is referred to Hersbach (2000).
Further details on some of the other scores discussed in sections 7.3 and
7.4 can also be found in Stanski et al. (1989).
7.5
Ensemble
statistics
An
ensemble, in general, is a collection of forecasts that sample the uncertainty
in the initial and forecast state of the system. At initial time, the ensemble
is centered at the best estimate of the state of the system (that can be
obtained either directly or by averaging an ensemble of analysis fields,
and will be called the control analysis). As discussed earlier, ensemble
forecasts are used widely for the generation of probabilistic forecasts.
The previous section was devoted to the verification of probabilistic forecasts
in general. Before ensemble forecasts are converted into probabilistic
information, however, it is desirable to evaluate their basic characteristics.
In
section 7.3 it was pointed out that the inherent value of forecast systems
lies in their ability to distinguish between cases when an event has a
higher or lower than climatological probability to occur in the future
(resolution), based on the initial conditions. As Figs. 7.3 and 7.4 demonstrate,
in chaotic systems resolution is limited in time. This is because in such
systems naturally occurring instabilities amplify initial and model related
uncertainties. Even though skill is reduced and eventually lost, forecasts
can remain (or can be calibrated to remain) statistically consistent with
observations (reliability).
An
ensemble forecast system that is statistically consistent with observations
is often called a perfect ensemble in a sense of perfect reliability. An
important property of a perfectly reliable ensemble is that the verifying
analysis (or observations) should be statistically indistinguishable from
the forecast members. Most of the verification tools specifically developed
and applied on ensemble forecasts are designed to evaluate the statistical
consistency of such forecasts. These additional measures of reliability,
as we will see below, can reveal considerably more detail as to the nature
and causes of statistically inconsistent behavior of ensemblebased probabilistic
forecasts than the reliability diagram (section 7.3.1) or the onedimensional
metric of the reliability component of the Brier score (section 7.3.2)
discussed above. By revealing the weak points of ensemble forecast systems,
the ensemblebased measures provide important information to the developers
of such systems that eventually can lead to improved probabilistic forecasts
as well.
7.5.1Ensemble
mean error and spread
If
the verifying analysis is statistically indistinguishable from the ensemble
members then its distance from the mean of the ensemble members (ensemble
mean error) must be statistically equal to the distance of the individual
members from their mean (ensemble standard deviation or spread, see, e.
g., Buizza 1997). Fig. 7.6 contrasts
the root mean square error (see chapter 5) and spread of the NCEP ensemble
mean forecast as a function of lead time. Initially, the ensemble spread
is larger than the ensemble mean error, indicating a larger than desired
initial spread. The growth of ensemble spread, however, is below that of
the error, which leads to insufficient spread at later lead times. This
is a behavior typical of current ensemble forecast systems that do not
properly account for model related uncertainties.
For
reliable forecast systems the ensemble mean error (that is equal to the
spread in the ensemble) can also be considered as a measure of resolution
(and therefore forecast skill in general). For example, an ensemble with
a lower average ensemble spread can more efficiently separate likely and
unlikely events from each other (and also has more forecast information
content). The ensemble mean forecast can also be compared to a single forecast
started from the same control analysis around which the initial ensemble
is centered (control forecast). Once nonlinearities become pronounced the
mean of an ensemble that properly describes the case dependent forecast
uncertainty should provide a better estimate of the future state of the
system than the control forecast (see Toth and Kalnay 1997) that generally
can fall anywhere within the cloud of the ensemble. In a properly formed
system ensemble mean error, thus, should be equal to or less than the corresponding
control forecast error, as seen in Fig. 7.6.
7.5.2
Equal likelihood histogram
Ensemble
forecast systems are designed to generate a finite set (m) of forecast
scenarios. Some ensemble forecast systems (e. g., those produced by ECMWF
and NCEP) use the same technique for generating each member of the ensemble
(i. e., same numerical prediction model, and same initial perturbation
generation technique). In other cases each ensemble member is generated
by a different model version (like in the ensemble at the Canadian Meteorological
Centre, see Houtekamer et al. 1996). In such cases individual ensemble
members may not perform equally well. Similarly, if the control forecast
is included in an otherwise symmetrically formed ensemble, the issue of
equal likelihood arises.
Whether
all ensemble members are equally likely or not is neither a desirable nor
an undesirable property of an ensemble in itself. When ensemble forecasts
are used to define forecast probabilities, however, one must know if all
ensemble members can be treated in a symmetric fashion. Whether all ensemble
members are equal can be tested by generating a histogram showing the number
of cases (accumulated over space and time) when each member is closest
to the verifying diagnostic (analyzed or observed state, see Zhu et al.
1996). Information from such a histogram can be useful as to how the various
ensemble members must be used in defining forecast probability values.
A flat (uneven) histogram, for example, indicates that all ensemble members
are (not) equally likely and can (cannot) be considered as independent
realizations of the same random process, indicating that the simple procedure
used in section 7.3.1 for converting ensemble forecasts into probabilistic
information is (not) applicable.
Fig.
7.7 contrasts the frequency of NCEP ensemble forecasts being best (i.
e., closest to the verification, averaged for 10 perturbed ensemble members)
with that of an equal and a higher resolution unperturbed control forecast
as a function of leadtime. Note first in Fig. 7.7 that the high resolution
control forecast, due to its ability to better represent nature, has an
advantage against the lower resolution members of the ensemble. This advantage,
however, is rather limited. As for the low resolution control forecast,
at short lead times, when the spread of the ensemble around the control
forecast is too large (see Fig. 7.6), it is somewhat more likely to be
closest to the verifying analysis. At longer lead times, when the spread
of the NCEP ensemble becomes deficient due to the lack of representation
of model related uncertainty, the control forecast becomes less likely
to verify best. When spread is too low, the ensemble members are clustered
too densely and the verifying analysis often lies outside of the cloud
of the ensemble. In this situation, since the control forecast is more
likely to be near the center of the ensemble cloud than the perturbed members,
a randomly chosen perturbed forecast has a higher chance of being closest
to the verifying observation than the control. The flat equal likelihood
histogram at intermediate lead times (indicated by the 48hour perturbed
and equal resolution control forecasts having the same likelihood in Fig.
7.7) thus is also an indicator of proper ensemble spread (cf. Fig. 7.6),
and hence close to perfect reliability.
7.5.3
Analysis rank histogram
If
all ensemble members are equally likely and statistically indistinguishable
from nature (i. e., the ensemble members and the verifying observation
are mutually independent realizations of the same probability distribution),
then each of the m+1 intervals defined by an ordered series of m
ensemble members, including the two open ended intervals, is equally likely
to contain the verifying value. Anderson (1996) and Talagrand et al. (1998)
suggested constructing a histogram by accumulating the number of cases
over space and time when the verifying analysis falls in any of the m+1
intervals. Such a graph is often referred to as the "analysis rank histogram".
Reliable
or statistically consistent ensemble forecasts lead to an analysis rank
histogram that is close to flat, indicating that each interval between
the ordered series of ensemble forecast values is equally likely (see 3day
panel in Fig. 7.8). An asymmetrical
distribution is usually an indication of a bias in the first moment of
the forecasts (see 15day lead time panel in Fig. 7.8) while a U (5day
panel in Fig. 7.8) or inverted Ushape (1day panel in Fig. 7.8) distribution
may be an indication of a positive or negative bias in the second moment
(variance) of the ensemble respectively. Current operational NWP ensemble
forecast systems in the medium leadtime range exhibit Ushaped analysis
rank histograms, which is an indication that more often than expected by
chance due to the finite ensemble size, the cloud of the ensemble misses
to encompass the verifying analysis.
7.5.4
Multivariate statistics
All
ensemble verification measures discussed so far are based on the accumulation
of statistics computed over single gridpoints. An ensemble system that
is found statistically consistent in this manner, however, will not necessarily
produce forecast patterns that are consistent with those observed (or analyzed).
Recently, various multivariate statistics have been proposed to evaluate
the statistical consistency of ensemble forecasts with analyzed fields
in a multivariate manner.
One
approach involves the computation of various statistics (like average distance
of each member from the other members) for a selected multivariate variable
(e. g., 500 hPa geopotential height defined over gridpoints covering a
preselected area), separately for cases when the verifying analysis is
included in, or excluded from the ensemble. A followup statistical
comparison of the two, inclusive and exclusive sets of statistics accumulated
over a spatiotemporal domain can reveal whether at a certain statistical
significance level the analysis can be considered part of the ensemble
in a multivariate sense (in case the two distributions are indistinguishable)
or not. Smith (2000) suggested the use of the nearest neighbor algorithm
for testing the statistical consistency of ensembles with respect to multivariate
variables in this fashion.
Another
approach is based on a comparison of forecast error patterns (e. g., control
forecast minus verifying analysis) and corresponding ensemble perturbation
patterns (control forecast minus perturbed forecasts). In a perfectly reliable
ensemble, the two sets of patterns are statistically indistinguishable.
The two sets of patterns can be compared either in a climatological fashion,
based, for example, on an empirical orthogonal function analysis of the
two sets of patterns over a large data set (for an example, see Molteni
and Buizza 1999), or on a case by case basis (see Wei and Toth 2002).
7.5.5
Time consistency histogram
The
concept of rank histograms can be used not only to test the reliability
of ensemble forecasts but also to evaluate the time consistency between
ensembles issued on consecutive days. Given a certain level of skill as
measured by the probabilistic scores discussed in section 7.3, an ensemble
system that exhibits less change from one issuing time to the next may
be of more value to some users. When constructing an analysis rank histogram,
in place of the verifying analysis one can use ensemble forecasts generated
at the next initial time. The "time consistency" histogram will assess
whether the more recent ensemble is a randomly chosen subset of the earlier
ensemble set.
Ideally,
one would like to see that with more information, newly issued ensembles
narrow the range of the possible, earlier indicated solutions, without
shifting the new ensemble into a range that has not been included in the
earlier forecast distribution. Such "jumps" in consecutive probabilistic
forecasts would result in a Ushaped time consistency histogram, indicating
suboptimal forecast performance. While control forecasts, representing
a single scenario within a large range of possible solutions, can exhibit
dramatic jumps from one initial time to the next, ensembles typically show
much less variations.
7.6Limitations
The
verification of probabilistic and ensemble systems, as that of any other
kind of forecasts, has its limitations. First, as pointed out earlier,
probabilistic forecasts can be evaluated only in a statistical sense. The
larger the sample size, the more stable and trustworthy the verification
results become. Given a certain sample size, one often needs to, or has
the option to subdivide the sample in search for more detailed information.
For example, when evaluating the reliability of continuoustype probability
forecasts one has to decide
when two forecast distributions are considered being the same. Grouping
more diverse forecast cases into the same category will increase sample
size but can potentially reduce useful forecast verification information.
Another
example concerns spatial aggregation of statistics. When the
analysis rank or other statistics is computed over large spatial or temporal
domains a flat histogram is a necessary but not sufficient condition for
reliability. Large and opposite local biases in the first and/or second
moments of the distribution may get canceled out when the local statistics
are integrated over larger domains (see, e. g., Atger 2002). In a careful
analysis, the conflicting demands for having a large sample to stabilize
statistics, and working with more specific samples (collected for more
narrowly defined cases, or over smaller areas) for gaining more insight
into the true behavior of a forecast system, need to be balanced.
So
far it has been implicitly assumed that observations are perfect. To some
degree this assumption is always violated. When the observational error
is comparable to the forecast errors observational uncertainty needs to
be explicitly dealt with in forecast evaluation statistics. A possible
solution is to introduce the same noise on the ensemble forecast values
as the observations have (Anderson 1996).
In
case of verifying ensemblebased forecasts, one should also consider the
effect of ensemble size. Clearly, a forecast based on a smaller ensemble
will provide a noisier and hence poorer representation of the underlying
processes, given the forecast system studied. Therefore special care should
be exercised when comparing ensembles of different sizes.
The
limitations described above must be taken into account not only in probabilistic
and ensemble verification studies, but also in forecast calibration where
probabilistic and/or ensemble forecasts are statistically postprocessed
based on forecast verification statistics.
7.7
Concluding remarks
In
this chapter we have reviewed various methods for the evaluation of probabilistic
and ensemble forecasts. Reliability and resolution were identified as the
two main attributes of probabilistic forecasts. Reliability is a measure
of statistical consistency between probabilistic forecasts and the corresponding
distribution of observations over the long run, while resolution measures
how different the probability values in perfectly reliable forecasts are
from the background (climatologically derived) probability values. The
ideal forecast system uses only 0 and 1 probability values and has a perfect
reliability. Note that this is a perfect dichotomous (nonprobabilistic)
forecast system, i. e., a deterministic forecast is a special case of probabilistic
forecasts, one without uncertainty.
In
the course of verifying probabilistic forecasts, their two main attributes:
reliability and resolution are assessed. Such a verification procedure,
just as that of any other type of forecasts, has its limitations. Most
importantly we recall that probabilistic forecasts can only be evaluated
on a statistical (and not individual) basis, preferably over a large sample.
When a stratification of all cases is warranted, a compromise has to be
found between the desire to learn more details about a forecast system
and the need for maintaining large enough subsamples to ensure the stability
of verification statistics. Additional limiting factors include the presence
of observational error, and to a lesser extent, the use of ensembles of
limited size. The issue of comparative verification, where two forecast
systems are intercompared, was also raised and the need for the use of
benchmark systems, against which a more sophisticated system can be compared,
was stressed.
It
was also pointed out that for temporally stationary forecast and observed
systems the reliability of forecasts can be made, at least theoretically,
perfect through a calibration procedure based on past verification statistics.
In contrast, resolution cannot be improved in such a simple manner. Thus
measuring the resolution of calibrated forecasts can provide sufficient
statistics for the evaluation of probabilistic forecasts.
The
relationship among the different verification scores like the Ranked Probability
Skill Score, the Relative Operating Characteristics, and the information
content that for perfectly calibrated probabilistic forecasts measures
the same attribute, resolution, is not clearly understood. Which scores
are best suited for certain applications is not clear either. It is important
to mention in this respect that the value of environmental forecasts can
also be assessed in the context of their use by society. The economic value
of forecasts has such significance that an entire chapter (Chapter 8) is
devoted to this subject. We close the present chapter by pointing out that
some of the verification scores discussed above have a clear link with
the economic value of forecasts. For example, the resolution component
of the Brier skill score, and the ROCarea, two measures of the resolution
of forecast systems, are equivalent to the economic value of forecasts
under certain assumptions(Murphy
1966; Richardson 2000).
6.
REFERENCES
Anderson,
J. L., 1996: A method for producing and evaluating probabilistic forecasts
from ensemble model integrations, J. Climate, 9, 15181530.
Atger,
F., 2001: Verification of intense precipitation forecasts from single models
and ensemble prediction systems. Nonlinear Processes in Geophysics, 8,401
 417.
Atger,
F., 2002: Spatial and interannual variability of the reliability of ensemble
based probabilistic forecasts: Consequences for calibration. Mon. Wea.
Rev., under review.
Bamber,
D., 1975, The area above the Ordinal Dominance Graph and the area Below
the Receiver Operating Characteristic Graph, Journal of Mathematical Psychology,
12, 387415.
Brier,
G. W., 1950, Verification of forecasts expressed in terms of probability,
Mon. Wea. Rev., 78, 13.
Buizza,
R., 1997: Potential forecast skill of ensemble prediction and spread and
skill distributions of the ECMWF ensemble prediction system. Mon. Wea.
Rev.,125, 99119.
Chatfield,
C., 2000: Time series forecasting. Chapman and Hall/CRC Press, p. 267.
Ehrendorfer,
M., 1997: Predicting the uncertainty of numerical weather forecasts: A
review. Meteorologische Zeitschrift, Neue Folge, 6, 147183.
Hersbach,
H., 2000: Decomposition of the continuous ranked probability score for
ensemble prediction systems. Weather and Forecasting, 15, 559570.
Houtekamer,
P. L., L. Lefaivre, J. Derome, H. Ritchie, and H. L. Mitchell, 1996: A
system simulation approach to ensemble prediction.Mon.
Wea. Rev., Mon. Wea. Rev., 124, 12251242.
Leith,
C. E., 1974: Theoretical skill of MonteCarlo forecasts.Mon.
Wea. Rev.,102, 409418.
Mason,
S. J., G. M. Mimmack, 2002: Comparison of Some Statistical Methods of Probabilistic
Forecasting of ENSO. Journal of Climate: Vol. 15, No. 1, pp. 829.
Molteni,
F., and Buizza, R., 1999: Validation of the ECMWF Ensemble Prediction System
using empirical orthogonal functions. Mon. Wea. Rev., 127, 23462358.
Molteni,
F., R. Buizza, T. N. Palmer, and T. Petroliagis, 1996:The
ECMWF ensemble system:Methodology
and validation.Q. J. R. Meteorol.
Soc., 122, 73119.
Murphy,
A. H., 1966: A note on the utility of probabilistic predictions and the
probability score in the costloss ratio decision situation. J. Appl. Meteor.,
5, 534537.
Murphy,
A. H., 1973, A new vector partition of the probability score, J. Appl.
Meteor., 12, 595600.
Richardson,
D. S., 2000: Skill and economic value of the ECMWF ensemble prediction
system,
Q.J.R.Meteorol.
Soc., 126, 649668.
Roulston,
M. S., and L. Smith, 2002: Evaluating Probabilistic Forecasts Using Information
Theory. Mon. Wea. Rev., 130, 16531660.
Smith,
L. A.,2000: Disentangling Uncertainty
and Error: On the Predictability of Nonlinear Systems. In Nonlinear Dynamics
and Statistics, ed. Alistair I. Mees, Boston, Birkhauser, 3164.
Stanski,
H. R., L. J. Wilson, and W. R. Burrows, 1989: Survey of common verification
methods in meteorology. WMO World Weather Watch Technical Reprot No. 8,
WMO/TD. No. 358.
Stevenson,
D. B., and F. J. DoblasReyes, 2000: Statistical methods for interpreting
Monte Carlo ensemble forecasts. Tellus, 52A, 300322.
Talagrand,
O., R. Vautard, and B. Strauss, 1998: Evaluation of probabilistic prediction
systems. Proceedings of ECMWF Workshop on Predictability, 2022 October
1997, 125.
Toth,
Z., and E. Kalnay, 1997: Ensemble forecasting at NCEP and the breeding
method.Mon.Wea.
Rev, 125, 32973319.
Toth,
Z., Y. Zhu, T. Marchok, S. Tracton, and E. Kalnay, 1998: Verification of
the NCEP global ensemble forecasts. Preprints of the 12th Conference on
Numerical Weather Prediction, 1116 January 1998, Phoenix, Arizona, 286289.
Wei,
M., and Toth, Z., 2002: Ensemble perturbations and forecast errors. Mon.
Wea. Rev., under review.
Zhu,
Y., G. lyengar, Z. Toth, M. S. Tracton, and T. Marchok, 1996: Objective
evaluation ofthe NCEP global ensemble forecasting system. Preprints of
the 15th AMS Conference on Weather Analysis and Forecasting, 1923 August
1996, Norfolk, Virginia, p. J79J82.
FIGURE
CAPTIONS
Figure
7.1. A hypothetical forecast probability density function p(x) (full
curve) for a onedimensional variable x, along with a verifying
observed value
x_{o} for a single case. The additional two
curves represent possible distributions for the verifying values observed
over a large number of cases when p(x) was forecast by two probabilistic
forecast systems. The distribution p_{1}(x) (dashdotted
curve) is close to the forecast distribution p(x), while the distribution
p_{2}(x) (dashed curve) is distinctly different from p(x)
(see text for discussion).
Figure
7.2. Reliability diagram for the NCEP Ensemble Forecast System (see text
for the definition of the event E under consideration). Full line: reliability
curve for the operational forecasts. Dashdotted line: reliability curve
for perfect ensemble forecasts where 'observation' is defined as one of
the ensemble members. The horizontal line shows the climatological frequency
of the event E,
p_{c} = 0.714. Insert in lower right: Sharpness graph (see
text for details).
Figure
7.3. Brier Skill Score (BSS_{c}, full curve, positively
oriented), and its reliability (B_{Rel}, short dash), and
resolution (B_{Res}, dashed, both negatively oriented) components,
as a function of forecast leadtime (days) for the ECMWF Ensemble Prediction
System (see text for the definition of the event E
under consideration).
Figure
7.4. ROC curves for the same event and predictions as in Figure 3, for
four different forecast ranges.
Figure
7.5. Information content as defined by Eq. 7.13 for calibrated probabilistic
forcecasts (with near perfect reliability) based on a 10member subset
of the NCEP ensemble. Forecasts are made for 10 climatologically equally
likely intervals for 500 hPa geopotential height values over the Northern
Hemisphere extratropics (2080 N), and are evaluated over the MarchMay
1997 period.
Figure
7.6. Root mean square error of 500 hPa geopotential height NCEP control
(open circle), ensemble mean (full circle), and climate mean (full square)
forecasts, along with ensemble spread (standard deviation of ensemble members
around their mean, open square), as a function of lead time, computed for
the Northern Hemisphere extratropics, averaged over December 2001  February
2002.
Figure
7.7. Equal likelihood diagram, showing the percentage of time when the
NCEP high (dashed) and equivalent resolution control (dashdotted), and
any one of the ten 0000 UTC perturbed ensemble 500 hPa geopotential height
forecasts (dotted) verify best out of a 23member ensemble (of which the
other 11 members are initialized 12 hours earlier, at 1200 UTC), accumulated
over gridpoints in the Northern Hemisphere extratropics during December
2001  February 2002. Chance expectation is 4.35 (solid).
Figure
7.8. Analysis rank histogram for a 10member 0000 UTC NCEP ensemble of
500 hPa geopotential height forecasts over the Northern Hemisphere extratropics
during December 2001  February 2002.