Issues:
Why verify?
Types of
forecasts and verification
What makes a forecast good?
Forecast quality vs. value
What is "truth"?
Validity of verification results
Pooling vs. stratifying results
Methods:
Standard verification
methods:
Methods for dichotomous (yes/no) forecasts
Methods for multicategory forecasts
Methods for forecasts of continuous variables
Methods
for probabilistic forecasts
Scientific or diagnostic
verification methods:
Methods
for spatial forecasts
Methods for
probabilistic forecasts, including ensemble prediction systems
Methods
for rare events
Other methods
Sample forecast datasets:
Finley
tornado forecasts
Probability of
precipitation forecasts
Some Frequently Asked Questions
References:
Links to other
verification
sites
References and further reading
Contributors to this site
The forecast is compared, or verified, against a corresponding observation of what actually occurred, or some good estimate of the true outcome. The verification can be qualitative ("does it look right?") or quantitative ("how accurate was it?"). In either case it should give you information about the nature of the forecast errors.
The three most important reasons to verify forecasts are:
Nature of forecast:  Example(s)  Verification methods 
deterministic (nonprobabilistic)  quantitative precipitation forecast  visual, dichotomous, multicategory, continuous, spatial 
probabilistic  probability of precipitation, ensemble forecast  visual, probabilistic, ensemble 
qualitative (worded)  5day outlook  visual, dichotomous, multicategory 
Spacetime domain: 

time series  daily maximum temperature forecasts for a city  visual, dichotomous, multicategory, continuous, probabilistic 
spatial distribution  map of geopotential height, rainfall chart  visual, dichotomous, multicategory, continuous, probabilistic, spatial, ensemble 
pooled space and time  monthly average global temperature anomaly  dichotomous, multicategory, continuous, probabilistic, ensemble 
Specificity of forecast: 

dichotomous (yes/no)  occurrence of fog  visual, dichotomous, probabilistic, spatial, ensemble 
multicategory  cold, normal, or warm conditions  visual, multicategory, probabilistic, spatial, ensemble 
continuous  maximum temperature  visual, continuous, probabilistic, spatial, ensemble 
object or eventoriented  tropical cyclone motion and intensity  visual, dichotomous, multicategory, continuous, probabilistic, spatial 
Consistency  the degree to which the forecast corresponds
to the forecaster's best judgement about the situation, based upon
his/her knowledge base
Quality  the degree to which the forecast corresponds
to what actually happened
Value  the degree to which the forecast helps the a
decision maker to realize some incremental economic and/or other
benefit
Since we're interested in forecast verification, let's look a bit closer at the forecast quality. Murphy described nine aspects (called "attributes") that contribute to the quality of a forecast. These are:
Bias  the correspondence between the mean forecast and mean observation.
Association  the strength of the linear relationship between
the forecasts and observations (for example, the correlation
coefficient
measures this linear relationship)
Accuracy  the level of agreement between the forecast and the
truth (as represented by observations). The difference between the
forecast
and the observation is the error. The lower the errors, the
greater
the accuracy.
Skill  the relative accuracy of the forecast over some
reference
forecast. The reference forecast is generally an unskilled forecast
such
as random chance, persistence (defined as the most recent set of
observations,
"persistence" implies no change in condition), or climatology. Skill
refers
to the increase in accuracy due purely to the "smarts" of the forecast
system. Weather forecasts may be more accurate simply because the
weather
is easier to forecast  skill takes this into account.
Reliability  the average agreement between the forecast values
and the observed values. If all forecasts are considered together, then
the overall reliability is the same as the bias. If the
forecasts
are stratified into different ranges or categories, then the
reliability
is the same as the conditional bias, i.e., it has a different
value
for each category.
Resolution  the ability of the forecast to sort or resolve
the set of events into subsets with different frequency distributions.
This means that the distribution of outcomes when "A" was forecast is
different
from the distribution of outcomes when "B" is forecast. Even if the
forecasts
are wrong, the forecast system has resolution if it can successfully
separate
one type of outcome from another.
Sharpness  the tendency of the forecast to predict extreme
values. To use a counterexample, a forecast of "climatology" has no
sharpness.
Sharpness is a property of the forecast only, and like resolution, a
forecast
can have this attribute even if it's wrong (in this case it would have
poor reliability).
Discrimination  ability of the forecast to discriminate among
observations, that is, to have a higher prediction frequency for an
outcome
whenever that outcome occurs.
Uncertainty  the variability of the observations. The greater
the uncertainty, the more difficult the forecast will tend to be.
Traditionally, forecast verification has emphasized accuracy and skill. It's important to note that the other attributes of forecast performance also have a strong influence on the value of the forecast.
Imagine a situation in which a high resolution numerical weather prediction model predicts the development of isolated thunderstorms in a particular region, and thunderstorms are indeed observed in the region but not in the particular spots suggested by the model. According to most standard verification measures this forecast would have poor quality, yet it might be very valuable to the forecaster in issuing a public weather forecast.
An example of a forecast with high quality but little value is a forecast of clear skies over the Sahara Desert during the dry season.
When the cost of a missed event is high, the deliberate overforecasting of a rare event may be justified, even though a large number of false alarms may also result. An example of such a circumstance is the occurence of fog at airports. In this case quadratic scoring rules (those involving squared errors) will tend to penalise such forecasts harshly, and a positively oriented score such as "hit rate" may be more useful.
Katz and Murphy (1997), Thornes
and Stephenson (2001) and Wilks (2001)
describe
methods for assessing the value of weather forecasts. The relative
value plot is sometimes used as a verification diagnostic.
In many cases it is difficult to know the exact truth because there are errors in the observations. Sources of uncertainty include random and bias errors in the measurements themselves, sampling error and other errors of representativeness, and analysis error when the observational data are analyzed or otherwise altered to match the scale of the forecast.
Rightly or wrongly, most of the time we ignore the errors in the
observational
data. We can get away with this if the errors in the observations are
much
smaller than the expected error in the forecast (high signal to noise
ratio).
Even skewed or undersampled verification data can give us a good idea
of which forecast products are better than others when
intercomparing
different forecast methods. Methods to account for errors in the
verification
data currently being researched.
The usual approach is to determine confidence intervals for the
verification
scores using analytic, approximate, or bootstrapping methods (depending
on the score). Some good meteorological references on this subject are
Seaman
et al. (1996), Wilks (2005, ch.5), Hamill
(1999), and Kane and Brown
(2000).
The danger with pooling samples, however, is that it can mask
variations
in forecast performance when the data are not homogeneous. It
can bias the results toward the most commonly sampled regime (for
example,
regions with higher station density, or days with no severe weather).
Nonhomegeneous samples can lead to overestimates of forecast skill
using some commonly used metrics  Hamill
and Juras (2005) provide some clear examples of how this can occur.
Stratifying the samples into quasihomogeneous subsets (by season, by geographical region, by intensity of the observations, etc.) helps to tease out forecast behavior in particular regimes. When doing this, be sure that the subsets contain enough samples to give trustworthy verification results.
One of the oldest and best verification methods is the good old
fashioned
visual, or "eyeball", method: look at the forecast and observations
side
by side and use human judgment to discern the forecast errors. Common
ways
to present data are as time series and maps.
The eyeball method is great if you only have a few forecasts, or you have lots of time, or you're not interested in quantitative verification statistics. Even when you do want statistics, it is a very good idea to look at the data from time to time!
However, the eyeball method is not quantitative, and it is very prone to individual, subjective biases of interpretation. Therefore it must be used with caution in any formal verification procedure.
The following sections give fairly brief descriptions of the standard verification methods and scores for dichotomous, multicategory, continuous, and probabilistic forecasts. For greater detail and discussion of the standard methods see Stanski et al. (1989) or one of the excellent books on forecast verification and statistics.
                                                                     
Methods for dichotomous (yes/no) forecasts
A dichotomous forecast says, "yes, an event will happen", or "no, the event will not happen". Rain and fog prediction are common examples of yes/no forecasts. For some applications a threshold may be specified to separate "yes" and "no", for example, winds greater than 50 knots.
To verify this type of forecast we start with a contingency table that shows the frequency of "yes" and "no" forecasts and occurrences. The four combinations of forecasts (yes or no) and observations (yes or no), called the joint distribution, are:
hit  event forecast to occur, and
did
occur
miss  event forecast not to occur,
but did occur
false alarm  event forecast to occur,
but did not occur
correct negative  event forecast not
to occur, and did not occur
The total numbers of observed and forecast occurrences and nonoccurences are given on the lower and right sides of the contingency table, and are called the marginal distribution.
Observed  
yes  no  Total  
Forecast  yes  hits  false alarms  forecast yes 
no  misses  correct negatives  forecast no  
Total  observed yes  observed no  total 
The contingency table is a useful way to see what types of errors are being made. A perfect forecast system would produce only hits and correct negatives, and no misses or false alarms.
A large variety of categorical statistics are computed from
the
elements in the contingency table to describe particular aspects of
forecast
performance. We will illustrate these statistics using a (madeup)
example.
Suppose a year's worth of official daily rain forecasts and
observations
produced the following contingency table:
Observed  
yes  no  Total  
Forecast  yes  82  38  120 
no  23  222  245  
Total  105  260  365 
Categorical statistics that can be computed from the yes/no contingency table are given below. Sometimes these scores are known by alternate names shown in parentheses.
Accuracy (fraction correct) 
Answers the question: Overall, what fraction of the forecasts were correct?
Range: 0 to 1. Perfect score: 1.
Characteristics: Simple, intuitive. Can be misleading since it is heavily influenced by the most common category, usually "no event" in the case of rare weather.
In the example above, Accuracy = (82+222) / 365 = 0.83, indicating that 83% of all forecasts were correct.
Bias score (frequency bias) 
Answers the question: How did the forecast frequency of "yes" events compare to the observed frequency of "yes" events?
Range: 0 to infinity. Perfect score: 1.
Characteristics: Measures the ratio of the frequency of forecast events to the frequency of observed events. Indicates whether the forecast system has a tendency to underforecast (BIAS<1) or overforecast (BIAS>1) events. Does not measure how well the forecast corresponds to the observations, only measures relative frequencies.
In the example above, BIAS = (82+38) / (82+23) = 1.14, indicating slight overforecasting of rain frequency.
Probability of detection (hit rate) 
Answers the question: What fraction of the observed "yes" events were correctly forecast?
Range: 0 to 1. Perfect score: 1.
Characteristics: Sensitive to hits, but ignores false alarms. Very sensitive to the climatological frequency of the event. Good for rare events.Can be artificially improved by issuing more "yes" forecasts to increase the number of hits. Should be used in conjunction with the false alarm ratio (below). POD is alsoan important component of the Relative Operating Characteristic (ROC) used widely for probabilistic forecasts.
In the example above, POD = 82 / (82+23) = 0.78, indicating that roughly 3/4 of the observed rain events were correctly predicted.
False alarm ratio 
Answers the question: What fraction of the predicted "yes" events actually did not occur (i.e., were false alarms)?
Range: 0 to 1. Perfect score: 0.
Characteristics: Sensitive to false alarms, but ignores misses. Very sensitive to the climatological frequency of the event. Should be used in conjunction with the probability of detection (above).
In the example above, FAR = 38 / (82+38) = 0.32, indicating that in roughly 1/3 of the forecast rain events, rain was not observed.
Probability of false detection (false alarm rate) 
Answers the question: What fraction of the observed "no" events were incorrectly forecast as "yes"?
Range: 0 to 1. Perfect score: 0.
Characteristics: Sensitive to false alarms, but ignores misses. Can be artificially improved by issuing fewer "yes" forecasts to reduce the number of false alarms. Not often reported for deterministic forecasts, but is an important component of the Relative Operating Characteristic (ROC) used widely for probabilistic forecasts.
In the example above, POFD = 38 / (222+38) = 0.15, indicating that for 15% of the observed "no rain" events the forecasts were incorrect.
Threat score (critical success index) 
Answers the question: How well did the forecast "yes" events correspond to the observed "yes" events?
Range: 0 to 1, 0 indicates no skill. Perfect score: 1.
Characteristics: Measures the fraction of observed and/or forecast events that were correctly predicted. It can be thought of as the accuracy when correct negatives have been removed from consideration, that is, TS is only concerned with forecasts that count. Sensitive to hits, penalizes both misses and false alarms. Does not distinguish source of forecast error. Depends on climatological frequency of events (poorer scores for rarer events) since some hits can occur purely due to random chance.
In the example above, TS = 82 / (82+23+38) = 0.57, meaning that slightly more than half of the "rain" events (observed and/or predicted) were correctly forecast.
Equitable threat score (Gilbert skill score) where
Answers the question: How well did the forecast "yes" events correspond to the observed "yes" events (accounting for hits due to chance)?
Range: 1/3 to 1, 0 indicates no skill. Perfect score: 1.
Characteristics: Measures the fraction of observed and/or forecast events that were correctly predicted, adjusted for hits associated with random chance (for example, it is easier to correctly forecast rain occurrence in a wet climate than in a dry climate). The ETS is often used in the verification of rainfall in NWP models because its "equitability" allows scores to be compared more fairly across different regimes. Sensitive to hits. Because it penalises both misses and false alarms in the same way, it does not distinguish the source of forecast error.
In the example above, ETS = (8234) / (82+23+3834) = 0.44. ETS gives a lower score than TS.
Hanssen and Kuipers discriminant (true skill statistic, Peirces's skill score) 
Answers the question: How well did the forecast separate the "yes" events from the "no" events?
Range: 1 to 1, 0 indicates no skill. Perfect score: 1.
Characteristics: Uses all elements in contingency table. Does not depend on climatological event frequency. The expression is identical to HK = POD  POFD, but the Hanssen and Kuipers score can also be interpreted as (accuracy for events) + (accuracy for nonevents)  1. For rare events HK is unduly weighted toward the first term (same as POD), so this score may be more useful for more frequent events. Can be expressed in a form similar to the ETS except the hits_{random} term is unbiased. See Woodcock (1976) for a comparison of HK with other scores.
In the example above, HK = 82 / (82+23)  38 / (38+222) = 0.63
Heidke skill score (Cohen's k)

where
Answers the question: What was the accuracy of the forecast relative to that of random chance?
Range: minus infinity to 1, 0 indicates no skill. Perfect score: 1.
Characteristics: Measures the fraction of correct forecasts after eliminating those forecasts which would be correct due purely to random chance. This is a form of the generalized skill score, where the score in the numerator is the number of correct forecasts, and the reference forecast in this case is random chance. In meteorology, at least, random chance is usually not the best forecast to compare to  it may be better to use climatology (longterm average value) or persistence (forecast = most recent observation, i.e., no change) or some other standard.
In the example above, HSS = 0.61
Odds ratio 
Answers the question: What is the ratio of the odds of a "yes" forecast being correct, to the odds of a "yes" forecast being wrong?
Odds ratio  Range: 0 to infinity, 1 indicates no skill. Perfect
score: infinity
Log odds ratio  Range: minus infinity to infinity, 0 indicates
no skill. Perfect score: infinity
Characteristics: Measures the ratio of the odds of making a
hit
to the odds of making a false alarm. The logarithm of the odds ratio is
often used instead of the original value. Takes prior probabilities
into
account. Gives better scores for rarer events. Less sensitive to
hedging.
Do not use if any of the cells in the contingency table are equal to
0.
Used widely in medicine but not yet in meteorology  see Stephenson
(2000) for more information.
Note that the odds ratio is not the same as the
ratio of the probability of making a hit (hits
/ # forecasts) to the probability of making a false
alarm
(false alarms / # forecasts),
since both of those can depend on the climatological frequency (i.e.,
the
prior probability) of the event.
In the example above, OR = (82 x 222) / (23 x 38) = 20.8, indicating that the odds of a "yes" prediction being correct are over 20 times greater than the odds of a "yes" forecast being incorrect.
Odds ratio skill score (Yule's Q) 
Answers the question: What was the improvement of the forecast over random chance?
Range: 1 to 1, 0 indicates no skill. Perfect score: 1
Characteristics: Independent of the marginal totals (i.e., of the threshold chosen to separate "yes" and "no"), so is difficult to hedge. See Stephenson (2000) for more information.
In the example above, ORSS = [(82 x 222)(23 x 38)] / [(82 x 222)+(23 x 38)] = 0.91
                                                                     
Methods for multicategory forecasts
Methods for verifying multicategory forecasts also start with
a contingency table showing the frequency of forecasts and observations
in the various bins. It is analogous to a scatter plot for categories.
Observed  Category  Total  
i,j  1  2  ...  K  
1  n(F_{1},O_{1})  n(F_{1},O_{2})  ...  n(F_{1},O_{K})  N(F_{1})  
Forecast  2  n(F_{2},O_{1})  n(F_{2},O_{2})  ...  n(F_{2},O_{K})  N(F_{2}) 
Category  ...  ...  ...  ...  ...  ... 
K  n(F_{K},O_{1})  n(F_{K},O_{2})  ...  n(F_{K},O_{K})  N(F_{K})  
Total  N(O_{1})  N(O_{2})  ...  N(O_{K})  N 
In this table n(F_{i},O_{j}) denotes the number of forecasts in category i that had observations in category j, N(F_{i}) denotes the total number of forecasts in category i, N(O_{j}) denotes the total number of observations in category j, and N is the total number of forecasts.
The distributions approach to forecast verification examines the relationship among the elements in the multicategory contingency table. A perfect forecast system would have values of nonzero elements only along the diagonal, and values of 0 for all entries off the diagonal. The offdiagonal elements give information about the specific nature of the forecast errors. The marginal distributions (N's at right and bottom of table) show whether the forecast produces the correct distribution of categorical values when compared to the observations. Murphy and Winkler (1987), Murphy et al. (1989) and Brooks and Doswell (1996) develop this approach in detail.
The advantage of the distributions approach is that the nature of the forecast errors can more easily be diagnosed. The disadvantage is that it is more difficult to condense the results into a single number. There are fewer statistics that summarize the performance of multicategory forecasts. However, any multicategory forecast verification can be converted to a series of K1 yes/notype verifications by defining "yes" to be "in category i" or "in category i or higher", and "no" to be "not in category i" or "below category i".
Histogram  Plot the relative frequencies of forecast and observed categories
Answers the question: How well did the distribution of forecast categories correspond to the distribution of observed categories?
Characteristics: Shows similarity between location, spread, and skewness of forecast and observed distributions. Does not give information on the correspondence between the forecasts and observations. Histograms give information similar to box plots.
          
Accuracy 
Answers the question: Overall, what fraction of the forecasts were in the correct category?
Range: 0 to 1. Perfect score: 1.
Characteristics: Simple, intuitive. Can be misleading since it is heavily influenced by the most common category.
Answers the question: What was the accuracy of the forecast in predicting the correct category, relative to that of random chance?
Range: minus infinity to 1, 0 indicates no skill. Perfect score: 1.
Characteristics: Measures the fraction of correct forecasts after eliminating those forecasts which would be correct due purely to random chance. This is one form of a generalized skill score, where the score in the numerator is the number of correct forecasts, and the reference forecast in this case is random chance. Requires a large sample size to make sure that the elements of the contingency table are all adequately sampled. In meteorology, at least, random chance is usually not the best forecast to compare to  it may be better to use climatology (longterm average value) or persistence (forecast is most recent observation, i.e., no change) or some other standard.
Hanssen and Kuipers discriminant (true skill statistic, Peirces's skill score) 
Answers the question: What was the accuracy of the forecast in predicting the correct category, relative to that of random chance?
Range: 1 to 1, 0 indicates no skill. Perfect score: 1
Characteristics: Similar to the Heidke skill score (above), except that in the denominator the fraction of correct forecasts due to random chance is for an unbiased forecast.
                                                                     
Methods for foreasts of continuous variables
Verifying forecasts of continuous variables measures how the
values
of the forecasts differ from the values of the observations. The
continuous
verification methods and statistics will be demonstrated on a sample
data
set of 10 temperature forecasts taken from Stanski
et al. (1989):
Day  1  2  3  4  5  6  7  8  9  10 
Forecast, F_{i }(C)  5  10  9  15  22  13  17  17  19  23 
Observation, O_{i} (C)  1  8  12  13  18  10  16  19  23  24 
Verification of continous forecasts often includes some exploratory plots such as scatter plots and box plots, as well as various summary scores.
Scatter plot  Plots the forecast values against the observed values.
Answers the question: How well did the forecast values correspond to the observed values?
Characteristics: Good first look at correspondence between forecast and observations. An accurate forecast will have points on or near the diagonal.
Scatter plots of the error can reveal relationships between the observed or forecast values and the errors.
Box plot  Plot boxes to show the range of data
falling
between the 25th and 75th percentiles, horizontal line inside the box
showing
the median value, and the whiskers showing the complete range of the
data.
Answers the question: How well did the distribution of forecast values correspond to the distribution of observed values?
Characteristics: Shows similarity between location, spread, and skewness of forecast and observed distributions. Does not give information on the correspondence between the forecasts and observations. Box plots give information similar to histograms.
Mean error 
Answers the question: What is the average forecast error?
Range: minus infinity to infinity. Perfect score: 0.
Characteristics: Simple, familiar. Also called the (additive) bias. Does not measure the magnitude of the errors. Does not measure the correspondence between forecasts and observations, i.e., it is possible to get a perfect score for a bad forecast if there are compensating errors.
In the example above, Mean Error = 0.8 C
(Multiplicative) bias 
Answers the question: How does the average forecast magnitude compare to the average observed magnitude?
Range: minus infinity to infinity. Perfect score: 1.
Characteristics: Simple, familiar. Best suited for quantities that have 0 as a lower or upper bound. Does not measure the magnitude of the errors. Does not measure the correspondence between forecasts and observations, i.e., it is possible to get a perfect score for a bad forecast if there are compensating errors.
In the example above, Bias = 1.06
Mean absolute error 
Answers the question: What is the average magnitude of the forecast errors?
Range: 0 to infinity. Perfect score: 0.
Characteristics: Simple, familiar. Does not indicate the direction of the deviations.
In the example above, MAE = 2.8 C
Root mean square error 
Answers the question: What is the average magnitude of the forecast errors?
Range: 0 to infinity. Perfect score: 0.
Characteristics: Simple, familiar. Measures "average" error, weighted according to the square of the error. Does not indicate the direction of the deviations. The RMSE puts greater influence on large errors than smaller errors, which may be a good things if large errors are especially undesirable, but may also encourage conservative forecasting.
In the example above, RMSE = 3.2 C
The root mean square factor is similar to RMSE, but gives a multiplicative error instead of an additive error.
Mean squared error 
Measures the mean squared difference between the forecasts and observations.
Range: 0 to infinity. Perfect score: 0.
Characteristics: Can be decomposed into component error sources following Murphy (1987). Units of MSE are the square of the basic units.
In the example above, MSE = 10 degrees squared
Linear error in probability space (LEPS)
Measures the error in probability space as opposed to measurement space, where CDFo() is the cumulative probability density function of the observations, determined from an appropriate climatology.
Range: 0 to 1. Perfect score: 0.
Characteristics: Does not discourage forecasting extreme values if they are warranted. Requires knowledge of climatological PDF. Not yet in wide usage  Potts et al. (1996) derived an improved version of the LEPS score that is equitable and does not "bend back" (give better scores for worse forecasts near the extremes):.
In the example above, suppose the climatological temperature is normally distributed with a mean of 14 C and variance of 50 C. Then according to the first expression, LEPS=0.106.
Correlation coefficient 
Addresses the question: How well did the forecast values correspond to the observed values?
Range: 1 to 1. Perfect score: 1.
Characteristics: Good measure of linear association or phase error. Visually, the correlation measures how close the points of a scatter plot are to a straight line. Does not take forecast bias into account  it is possible for a forecast with large errors to still have a good correlation coefficient with the observations. Sensitive to outliers.
In the example above, r = 0.914
Anomaly correlation 
Addresses the question: How well did the forecast anomalies correspond to the observed anomalies?
Range: 1 to 1. Perfect score: 1.
Characteristics: Measures correspondence or phase difference between forecast and observations, subtracting out the climatological mean at each point, C, rather than the sample mean values. The anomaly correlation is frequently used to verify output from numerical weather prediction (NWP) models. AC is not sensitive to forecast bias, so a good anomaly correlation does not guarantee accurate forecasts. Both forms of the equation are in common use  see Jolliffe and Stephenson (2003) or Wilks (2005) for further discussion.
In the example above, if the climatological temperature is 14 C, then AC = 0.904. AC is more often used in spatial verification.
S1 score 
where DF
(DO)
refers to the horizontal gradient in the forecast (observations).
Answers the question: How well did the forecast gradients correspond to the observed gradients?
Range: 0 to infinity. Perfect score: 0.
Characteristics: It is usually applied to geopotential height or sea level pressure fields in meteorology. Long historical records in NWP showing improvement in model performance over the years. Because S1 depends only on gradients, good scores can be achieved even when the forecast values are biased. Also depends on spatial resolution of the forecast.
Answers the question: What is the relative improvement of the forecast over some reference forecast?
Range: Lower bound depends on what score is being used to compute skill and what reference forecast is used, but upper bound is always 1; 0 indicates no improvement over the reference forecast. Perfect score: 1.
Characteristics: Implies information about the value or worth of a forecast relative to an alternative (reference) forecast. In meteorology the reference forecast is usually persistence (no change from most recent observation) or climatology. The skill score can be unstable for small sample sizes. When MSE is the score used in the above expression then the resulting statistic is called the reduction of variance.
          
See also Methods for spatial forecasts for more scientific/diagnostic techniques.
                                                                     
Methods for probabilistic forecasts
A probabilistic forecast gives a probability of an event occurring, with a value between 0 and 1 (or 0 and 100%). In general, it is difficult to verify a single probabilistic forecast. Instead, a set of probabilistic forecasts, p_{i}, is verified using observations that those events either occurred (o_{i}=1) or did not occur (o_{i}=0).
An accurate probability forecast system has:
reliability  agreement between
forecast
probability and mean observed frequency
sharpness  tendency to forecast
probabilities
near 0 or 1, as opposed to values clustered around the mean
resolution  ability of the forecast
to resolve the set of sample events into subsets with
characteristically
different outcomes
Reliability diagram  (called "attributes diagram" when the noresoloution and noskill w.r.t. climatology lines are included).
The reliability diagram plots the observed frequency against the forecast probability, where the range of forecast probabilities is divided into K bins (for example, 05%, 515%, 1525%, etc.). The sample size in each bin is often included as a histogram or values beside the data points.
Answers the question: How well do the predicted probabilities of an event correspond to their observed frequencies?
Characteristics: Reliability is indicated by the proximity of
the plotted curve to the diagonal. The deviation from the diagonal
gives
the conditional bias. If the curve lies below the line, this
indicates
overforecasting (probabilities too high); points above the line
indicate
underforecasting (probabilities too low). The flatter the curve in the
reliability diagram, the less resolution it has. A forecast of
climatology
does not discriminate at all between events and nonevents, and thus
has
no resolution. Points between the "no skill" line and the diagonal
contribute
positively to the Brier skill score. The frequency
of
forecasts in each probability bin (shown in the histogram) shows the
sharpness
of the forecast.
The reliability diagram is conditioned on the forecasts (i.e., given
that X was predicted, what was the outcome?), and can be expected to
give
information on the real meaning of the forecast. It it a good partner
to
the ROC, which is conditioned on the observations.
Brier score 
Answers the question: What is the magnitude of the probability forecast errors?
Measures the mean squared probability error. Murphy (1973) showed that it could be partitioned into three terms: (1) reliability, (2) resolution, and (3) uncertainty.
Range: 0 to 1. Perfect score: 0.
Characteristics: Sensitive to climatological frequency of the event: the more rare an event, the easier it is to get a good BS without having any real skill. Negative orientation (smaller score better)  can "fix" by subtracting BS from 1.
Brier skill score 
Answers the question: What is the relative skill of the
probabilistic
forecast over that of climatology, in terms of predicting whether or
not
an event
occurred?
Range: minus infinity to 1, 0 indicates no skill when compared to the reference forecast. Perfect score: 1.
Characteristics: Measures the improvement of the probabilistic forecast relative to a reference forecast (usually the longterm or sample climatology), thus taking climatological frequency into account. Not strictly proper. Unstable when applied to small data sets; the rarer the event, the larger the number of samples needed.
Relative operating characteristic Plot hit rate (POD) vs false alarm rate (POFD), using a set of increasing probability thresholds (for example, 0.05, 0.15, 0.25, etc.) to make the yes/no decision. The area under the ROC curve is frequently used as a score.
Answers the question: What is the ability of the forecast to discriminate between events and nonevents?
ROC: Perfect: Curve travels from bottom left to top
left
of diagram, then across to top right of diagram. Diagonal line
indicates
no skill.
ROC area: Range: 0 to 1, 0.5 indicates no skill.
Perfect
score: 1
Characteristics: ROC measures the ability of the forecast to
discriminate between two alternative outcomes, thus measuring
resolution.
It is not sensitive to bias in the forecast, so says nothing about
reliability.
A biased forecast may still have good resolution and produce a good ROC
curve, which means that it may be possible to improve the forecast
through
calibration. The ROC can thus be considered as a measure of potential
usefulness.
The ROC is conditioned on the observations (i.e., given that Y
occurred,
what was the correponding forecast?) It is therefore a good
companion
to the reliability diagram, which is
conditioned
on the forecasts.
More information on ROC can be found in Mason
1982, Jolliffe and
Stephenson
2003 (ch.3), and the WISE
site.
Ranked probability score 
where M is the number of forecast categories,
p_{k}
is the predicted probability in forecast category k, and o_{k}
is an indicator (0=no, 1=yes) for the observation in category k.
Answers the question: How well did the probability forecast predict the category that the observation fell into?
Range: 0 to 1. Perfect score: 0.
Characteristics: Measures the sum of squared differences in
cumulative
probability space for a multicategory probabilistic forecast.
Penalizes
forecasts more severely when their probabilities are further from the
actual
outcome. Negative orientation  can "fix" by subtracting
RPS from
1. For two forecast categories the RPS is the same as the Brier
Score.
Continuous version 
Ranked probability skill score 
Answers the question: What is the relative improvement of the probability forecast over climatology in predicting the category that the observations fell into?
Range: minus infinity to 1, 0 indicates no skill when compared to the reference forecast. Perfect score: 1.
Characteristics: Measures the improvement of the multicategory probabilistic forecast relative to a reference forecast (usually the longterm or sample climatology). Strictly proper. Takes climatological frequency into account. Unstable when applied to small data sets.
Relative value (value score) (Richardson,
2000; Wilks, 2001)
Answers the question: For a cost/loss ratio C/L for taking action based on a forecast, what is the relative improvement in economic value between climatalogical and perfect information?
Range: minus infinity to 1. Perfect score: 1.
Characteristics: The relative value is a skill score of expected expense, with climatology as the reference forecast. Because the cost/loss ratio is different for different users of forecasts, the value is generally plotted as a function of C/L.
Like ROC, it gives information that can be used in decision making. When applied to a probabilistic forecasts system (for example, an ensemble prediction system), the optimal value for a given C/L may be achieved by a different forecast probability threshold than the optimal value for a different C/L. In this case it is necessary to compute relative value curves for the entire range of probabilities, then select the optimal values (the upper envelope of the relative value curves) to represent the value of the probabilistic forecast system. Click here for more information on the cost/loss model and relative value.
          
See also Methods for ensemble
prediction
systems for more scientific/diagnostic techniques.
Scientific or diagnostic verification methods
Scientific, or diagnostic, verification methods delve more deeply into the nature of forecast errors. As a result they are frequently more complex than the standard verification measures described earlier. Distributionsoriented approaches and plots such as histograms, box plots, and scatter plots, are standard diagnostic verification methods.
This section gives very brief descriptions of several recently developed scientific and diagnostic methods, and relies heavily on references and links to other sites with greater detail.
This is also a place to promote new verification techniques. If you are working in this area, then you are encouraged to share your methods via this web site.
                                                                     
Scale decomposition methods  allow the errors at each scale to be diagnosed:
Wavelet decomposition (Briggs
and Levine, 1997)
          
Intensityscale
verification approach (Casati et al.
2004)
Answers
the question: How does the skill
of spatial precipitation forecasts depend on both the scale of the
forecast error and the intensity of the precipitation events?
The intensityscale verification approach bridges
traditional categorical binary verification, which provides information
about skill for different precipitation intensities, with the more
recent techniques which evaluate the forecast skill on different
spatial scales (e.g., ZepedaArce et al., 2000; Briggs
and Levine, 1997). It assesses the forecast on its whole domain, and is
well suited for verifying spatially discontinuous fields, such as
precipitation fields characterized by the presence of many scattered
precipitation events. It provides useful insight on
individual forecast cases as well as for forecast systems evaluated
over many cases.
Forecasts are assessed using the Mean Squared Error
(MSE) skill score of binary images, obtained from the forecasts and
analyses by thresholding at different precipitation rate intensities.
The skill score is decomposed on different spatial scales using a
twodimensional discrete Haar wavelet decomposition of binary error
images. The
forecast skill can then be evaluated in terms of precipitation rate
intensity and spatial scale.
                                                                     
Discrete cosine transformation (DCT) (Denis et al., 2002a for method; Denis et al., 2002b and de Elia et al., 2002 for application)
                                 
                                   
Fuzzy (neighborhood) methods  relax the
requirement for an exact match by evaluating forecasts
in the local neighborhood of the observations.
Multiscale statistical organization (ZepedaArce et al., 2000)
                                 
                                   
Fractions skill score (Roberts and Lean, 2008)
Answers the question: What are the spatial scales at which the forecast resembles the observations?
This approach directly compares the
forecast and observed
fractional coverage of gridbox events (rain exceeding a certain
threshold, for example) in spatial windows of increasing size. These
event
frequencies are used directly to compute a Fractions Brier Score,
a version of the more familiar (half) Brier score
but now the
observation can take any value between 0 and 1. The result can be
framed as a Fractions Skill Score with the following properties:
                          
      
                                   
Fuzzy logic (Damrath, 2004)
                          
      
                                   
Pragmatic (neighborhood) method (Theis et al., 2005)
                                                                     
Spatial
multievent
contingency
tables  useful for verifying high resolution forecasts (Atger,
2001).
By using multiple thresholds, a deterministic
forecast system can be
evaluated across a range of possible decision thresholds
(instead of just
one) using ROC
and relative
value. The decision thresholds might be intensity thresholds or
even "closeness"
thresholds (for example, forecast event within 10 km of the location of
interest, within 20 km, 30 km, etc.). Such verification results can
be used to assess the performance of high resolution forecasts
where the exact spatial matching of forecast and observed events is
difficult
or unimportant. This multithreshold approach enables a fairer
comparison against ensemble prediction
systems or other probabilistic forecasts.
(related work: Tremblay et al., 1996)
                                                                     
Practically perfect hindcasts  assessing relative skill of spatial forecasts (Brooks et al, 1998; Kay, 2002)
                                                                     
Fuzzy verification
framework  12 fuzzy (neighborhood) methods combined into one
framework (Ebert, 2008)
"Fuzzy" verification approaches reward closeness by relaxing the requirement for exact matches between forecasts and observations. Some of these fuzzy methods compute standard verification metrics for deterministic forecasts using a broader definition of what constitutes a "hit". Other fuzzy methods treat the forecasts and/or observations as probability distributions and use verification metrics suitable for probability forecasts. Implicit in each fuzzy verification method is a particular decision model concerning what constitutes a good forecast.
                                 
                                   
CRA (entitybased) verification (Beth Ebert, Bureau of Meteorology, Australia)
Answers the question:
What is the location
error of the (spatial) forecast, and how does the total error break down
into components due to incorrect location, volume, and fine scale
structure?
This objectoriented method verifies the properties of spatial forecasts of
entities,
where an entity is anything that can be defined by a closed contour. Some
examples of entities, or blobs, are contiguous rain areas (CRAs, for which
the method is named), convective outlook regions, and low pressure minima.
For each entity that can be identified in the forecast and the observations,
CRA verification uses pattern matching techniques to determine the location
error, as well as errors in area, mean and maximum intensity, and spatial
pattern. The total error can be decomposed into components due to location,
volume, and pattern error. This is a useful property for model developers
who need such information to improve the numerical weather prediction
models.
In addition, the verified entities themselves may be classified as "hits", "misses", etc., according to how close the forecast location was to the observed location, and how well the maximum intensity was represented by the forecast. This event verification can be useful for monitoring forecast performance.
Click here to learn more, or see Ebert and McBride (2000).
                                                                     
Method for Objectbased Diagnostic
Evaluation (MODE) (Brown
et al.,
2004; Davis et al., 2006)
Answers the question: How similar are the forecast objects to the observed objects according to a variety of descriptive criteria ?
MODE uses a convolution filter and thresholding to first identify objects in gridded fields. Performance at different spatial scales can be investigated by varying the values of the filter and threshold parameters. Then a fuzzy logic scheme is used to merge objects within a field, and match them between the forecast and the observations. Several attributes of the matched objects (location, area, volume, intensity, shape, etc.) are compared to see how similar they are. These are combined to give an "interest value" that summarizes the goodness of the match.
Output of the MODE algorithm include:
Attributes can be summarized across many cases to understand how forecasts represent the storm/precipitation climatology, understand systematic errors, and document variability in performance in different situations.
The MODE verification scheme is part of the Model Evaluation Tools (MET) toolkit freely available from NCAR. More information on MODE is available from the Developmental Testbed Center.
                                                                     
Objectoriented verification ideas (Baldwin et al, 2002)
                                                                     
Event verification using composites (Nachamkin, 2004)
                                                                     
Cluster analysis (Marzban and Sandgathe, 2006, 2008)
                                                                     
Procrustes shape analysis (Michaes et al., 2007)
                                                                     
StructureAmplitudeLocation
(SAL) method (Wernli et al., 2008)
                                                                     
Automated eastwest phase error calculation (Keith Brill, NOAA/NWS/NCEP/HPC)
Answers the question: What is the phase error of the (spatial) forecast?
This approach considers both high and low pressure centers, troughs, and ridges, and takes into account the typical synoptic scale wavelength.
Gridded forecasts and analyses of mean sea level pressure are meridionally averaged within a zonal strip to give an eastwest series of forecast and analyzed values. Cosine series trigonometric approximations are applied to both series, and the variance associated with each spectral component is computed. These are then sorted in descending order of variance to get the hierarchy of most important waves. If the hierarchies agree between the forecast and analyzed spectral components, then the phase angle (error) can be computed for each component.
In practice, the first spectral component is usually responsible for most of the variance and is the main one of interest. The phase errors are presented as time series. Click here to learn more.
                                                                     
Feature calibration and alignment (Hoffman et al., 1995; Nehrkorn et al., 2003)
                                                                     
                                                                     
Multicategory reliability diagram (Hamill, 1997)
                                                                     
Rank histogram (Talagrand et al, 1997; Hamill, 2001)
Answers the question: How well does the ensemble spread of the forecast represent the true variability (uncertainty) of the observations?
Also known as a "Talagrand diagram", this method checks where the verifying observation usually falls with respect to the ensemble forecast data, which is arranged in increasing order at each grid point. In an ensemble with perfect spread, each member represents an equally likely scenario, so the observation is equally likely to fall between any two members.
To construct a rank histogram, do the following:
1. At every observation (or analysis) point rank the N ensemble
members from lowest to highest. This represents N+1 possible bins that
the observation could fit into, including the two extremes
2. Identify which bin the observation falls into at each point
3. Tally over many observations to create a histogram of rank.
Interpretation:
Flat  ensemble spread about right to represent forecast uncertainty
Ushaped  ensemble spread too small, many observations falling outside
the extremes of the ensemble
Domeshaped  ensemble spread too large, most observations falling
near the center of the ensemble
Asymmetric  ensemble contains bias
Note: A flat rank histogram does not necessarily indicate a good forecast, it only measures whether the observed probability distribution is well represented by the ensemble.
                                                                     
Correspondence
ratio
 ratio
of the area of intersection of two or more events to the combined area
of those events (Stensrud
and Wandishin, 2000)
where F_{m,i} is the value of forecast m at gridpoint
i,
and O_{i} is the corresponding observed value. In the diagram
CR
is the ratio of the dark area to the total shaded area. Click
here
to learn more.
                                                                     
Likelihood skill measure  Likelihood is defined very simply as the probability of the observations given the forecast. Likelihoodbased measures can be used for binary and continuous probability forecasts, and provide a simple and natural general framework for the evaluation of all kinds of probabilistic forecasts. For more information see Jewson, (2003)
                                 
                                   
Logarithmic
scoring rule (ignorance
score) (Roulston and Smith,
2002)
The logarithmic scoring rule can be defined as follows: If there are n (mutually exclusive) possible outcomes and f_{i} (i=1,...n) is the predicted probability of the i^{th} outcome occurring then if the j^{th} outcome is the one which actually occurs the score for this particular forecastrealization pair is given by
                                                                     
Deterministic limit (Hewson, 2007)
The 'deterministic limit' is defined, for categorical forecasts of a predefined rare meteorological event, to simply be the point ahead of issue time at which, across the population, the number of misses plus false alarms equals the number of hits (i.e. critical success index =0.5). A hypothetical example of an accuracy statement that might thus arise would be: 'The deterministic limit for predicting a windstorm, with gusts in excess of 60 kts at one or more lowlying inland stations in NW Europe, is 2.1 days'. The base rate (or event frequency) should also be disclosed. Recalibration of the forecast is often necessary for useful deterministic limit measures to be realised.
As they provide a clear measure of capability, deterministic limit values for various parameters may in due course be used as yearonyear performance indicators, and also to provide succinct guidelines for warning service provision. They could also be used as the cutoff point to switch from deterministic to probabilistic guidance. In turn this may help elevate the hitherto muted enthusiasm shown, by some customers, for probabilistic forecasts.
Click here
to learn more.
                                                                     
Extreme dependency score 
Answers the question: What is the association between forecast and observed rare events?
Range: 1 to 1, 0 indicates no skill. Perfect score: 1
Characteristics: Converges to 2η1 as event frequency approaches 0, where η is a parameter describing how fast the hit rate converges to zero for rarer events. EDS is independent of bias, so should be presented together with the frequency bias. See Stephenson et al (2008) for more information.
                                                                     
Probability model approach (Ferro, 2007)  Probability models that impose parametric forms on the relationships between observations and forecasts can help to quantify forecast quality for rare, binary events by identifying key features of the relationships and reducing sampling variation of verification measures. Click here to learn more.
                                                                     
                                                                     
Root mean squared factor
(Golding,
1998)
Answers the question: What is the average multiplicative
error?
The RMSF is the exponent of the root mean square error of the
logarithm of the data. The logarithmic transformation is performed to
smooth the
data, reduce the discontinuities, and make the data more robust.
Whereas the RMS error can be interpreted as giving a scale to the additive
error, i.e., f =
o +/ RMS, the RMSF
can be interpreted as giving a scale to the multiplicative error, i.e.,
f =
o *// RMSF (read: "multiplied or divided
by"), which is a more appropriate measure of accuracy for some variables
and more intuitively meaningful than the RMS log error. In order to
avoid assigning skill to trivial forecasts, statistics are only accumulated
where either the forecast or observations are within specified limits. For
example, for visibility verification, the lower and upper limits used by Golding
(1998) were 1 m and 5000 m. When either the forecast or the observation
lies within the range but the other is outside the range, then limits
of half the lower limit or double the upper limit are prescribed on the other.
                                                                     
Quantilebased categorical statistics (Jenkner et al., 2008)
Click here to learn more.
Sample forecast datasets
In March 1884 Sergeant John Finley initiated twice daily tornado forecasts for eighteen regions in the United States, east of the Rocky Mountains. Finley claimed 95.6% to 98.6% overall accuracy for the first 3month period, depending on the time and district, with some districts achieving 100% accuracy for all 3 months. A critic of the results pointed out that 98.2% accuracy could be had by merely forecasting "no tornado"! This clearly illustrates the need for more meaningful verification scores.
The contingency table for Finley's (1884) forecasts is:
Observed  
tornado  no tornado  Total  
Forecast  tornado  28  72  100 
no tornado  23  2680  2703  
Total  51  2752  2803 
Click here
to see how the
different categorical scores rate the Finley (1884) forecasts.
2. What is the best statistic for measuring the accuracy of a forecast?
3. Why, when a model's resolution is improved, do the forecasts often verify worse?
5. How do I verify worded forecasts?
6. What does "hedging" a forecast mean, and how do some scores encourage hedging?
7. Is there a difference between "verification" and "validation"?
8. What is the relationship between confidence intervals and prediction intervals?
9.
How do I know whether one forecast
system performs significantly better than another?
10.
What are the challenges and strategies to verify weather and climate extremes?
...
Statistics
The R Project for Statistical
Computing  free software for statistical computing and graphics.
Click here
to find out how to get the R forecast verification routines.
A New View of
Statistics
 Will Hopkins' statistical primer for the health sciences
Engineering
Statistics Handbook  NIST / SEMATECH summaries of statistical
methods
Web Interface for
Statistics Education (WISE)  teaching resources offered through
Introductory Statistics courses, especially in the social sciences
Dr. Arsham's Web
Page  zillions of links to webbased statistics resources
Meteorological 
examples
NOAA Forecast
Systems Laboratory's (FSL) Real Time Verification System (RTVS) 
large variety of realtime verification results with an aviation emphasis
Verification of
NCEP model QPFs  rain maps and verification scores for regional
and mesoscale models over the USA
MOS Verification over
the US  operational verification of temperature and probability of
precipitation forecasts using several scores
Ensemble
Evaluation and Verification  NCEP ensemble prediction system verification
DEMETER
Verification  deterministic and probabilistic verification of EU
multimodel ensemble system for seasonal to interannual prediction
Workshops
4th International Verification Methods Workshop, 810 June 2009, Helsinki, Finland 
Presentations and tutorial lectures.
3rd
International Verification Methods Workshop, 31 January2 February, 2007,
Reading,UK  Tutorial lecture notes and scientific presentations. Click
here
to see the 2008 special issue of Meteorological Applications on
Forecast Verification that features papers from the workshop.
2nd
International Verification Methods Workshop, September 1517, 2004, Montreal,
Canada  Presentations and discussion
Workshop
on Making Verification More Meaningful, Boulder, CO, 30 July  1
August 2002  Presentations and posters
RFC
River Forecast Verification Workshop, Silver Spring, MD, 2728
February 2002  Talks on verification of river forecasts
WWRP/WMO
Workshop on the Verification of Quantitative Precipitation Forecasts,
Prague, Czech Republic, 1416 May 2001  papers on verification of QPFs
SRNWP
Mesoscale Verification Workshop 2001, KNMI, De Bilt, The
Netherlands, 2324 April 2001
Katz, R.W. and A.H. Murphy (eds), 1997: Economic Value of Weather and Climate Forecasts. Cambridge University Press, Cambridge.
Jolliffe, I.T., and D.B. Stephenson, 2003: Forecast Verification. A Practitioner's Guide in Atmospheric Science. Wiley and Sons Ltd, 240 pp. Click here to see the Table of Contents.
Murphy, A.H. and R.W. Katz, ed., 1985: Probability, Statistics, and Decision Making in the Atmospheric Sciences. Westview Press, Boulder, CO.
Nurmi, P., 2003: Recommendations on the verification of local weather forecasts (at ECWMF member states). ECMWF Operations Department, October 2003. Click here to access a PDF version (464 kB).
Stanski, H.R., L.J. Wilson, and W.R. Burrows, 1989: Survey of common verification methods in meteorology. World Weather Watch Tech. Rept. No.8, WMO/TD No.358, WMO, Geneva, 114 pp. Click here to access a PDF version.
von Storch, H. and F.W. Zwiers, 1999: Statistical Analysis in Climate Research. Cambridge University Press, Cambridge.
Wilks, D.S., 2005: Statistical Methods in the Atmospheric Sciences. 2nd Edition. Elsevier, 627 pp.
Special issue of Meteorological Applications (2008) on Forecast Verification
(See also Harold Brooks' site for an extensive reference list)
Atger, F., 2001:
Verification of intense precipitation
forecasts from single models and ensemble prediction systems. Nonlin.
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Atger, F., 2003: Spatial and interannual variability of the
reliability of ensemblebased probabilistic forecasts: Consequences for
calibrations. Mon. Wea. Rev., 131, 15091523.
Atger, F., 2004: Relative impact of model quality and ensemble
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Frédéric Atger, MétéoFrance, Toulouse, France
Mike Baldwin, Purdue University, West Lafayette, Indiana
Keith Brill, NOAA/NWS Hydrometeorological Prediction Center, Washington DC
Harold Brooks, National Severe Storms Laboratory, Norman, Oklahoma
Barb Brown, NCAR, Boulder, Colorado
Barbara Casati, Ouranos, Montreal, Canada
Ulrich Damrath, Deutscher Wetterdienst, Offenbach, Germany
Beth Ebert, Bureau of Meteorology, Melbourne, Australia
Anna Ghelli, ECMWF, Reading, UK
Martin Göber, Deutscher Wetterdienst, Offenbach, Germany
Johannes Jenkner, International Research Institute for Climate Prediction
Ian Jolliffe, University of Aberdeen
Pertti Nurmi, Finnish Meteorological Institute, Hensinki, Finland
David Stephenson, University of Exeter, UK
Clive Wilson, The Met Office, Bracknell, UK
Laurie Wilson, Research en Prévision Numérique, Dorval, Canada
Chris Ferro, University of Exeter, UK
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