Understanding the "Probability of Exceedance" Forecast Graphs


Purpose of the graphs:

The "probability of exceedance" curves give the forecast probability that a temperature or precipitation quantity, shown on the horizontal axis, will be exceeded at the location in question, for the given season at the given lead time. The information on these graphs is consistent with the information given in the forecast maps of probability anomaly that have been issued since the beginning of 1995. Those forecast maps show the probability anomaly of the most favored tercile of the climatological distribution: below normal, near normal, or above normal. The graphs shown here, on the other hand, are intended to provide additional detail about the forecast probability distribution at an individual location–i.e., any one of 102 climate regions in the mainland U.S., or an individual station in Alaska, Hawaii or a Pacific island. The additional information comes about through the display of the entire probability distribution, as opposed to just the probability anomaly of the most! favored tercile. With the entire distribution, users may select any cutoffs or categories that are of particular interest, and are not limited to a pre-established tercile category. Although skill in climate forecasting is in most cases modest in absolute terms, there is nonetheless justification to issue a complete forecast probability distribution. Such a distribution is an attempt to accurately convey the sense of the forecast while also conveying the degree of uncertainty (which is often very high) contained in that forecast. In an additional new web site facility, a flexible user-prescribed probability assessment is provided so that a user can use the "probability of exceedance" curve to automatically determine the forecast probability of occurrence with respect to their own upper and lower limits.

What the curves mean:

Each graph contains four curves.

The first curve, shown in black, shows the "normal", or climatological probability distribution. It is derived by computing the average, and also computing a measure of the degree of year-to-year variation around that average. The curve is therefore called a "fitted" curve, because it is defined using a formula that makes it possible to construct a smooth curve to the data. The data may not be so smooth and regular, but the formula only uses the average and the typical deviations from that average to define the curve. The center (such as the mean or median) of this distribution is based on the historical record of observations at the given location and season during the period used as the normal based period. For example, in 1999 the normal base period is 1961-90. Upon reaching the year 2001, the base period is expected to be updated to 1971-2000. The value of the center of the distribution, or normal, is printed numerically near the top of the graph. The variability, or sprea! d, of the climatological distribution is based on a period longer than the normal base period. This is done in order to obtain a more accurate estimate of the variability. Getting an acceptably accurate estimate of the variability requires more cases than getting an acceptably accurate estimate of the center of the distribution. Some of the extra years used in getting the variability estimate fall immediately prior to the base period, while others fall immediately following it, depending on how many years exist between the end of the base period and the present. For temperature, the value of the center of the distribution number represents the temperature at which the curve crosses the 50% point on the vertical axis. For precipitation, whose distribution is not symmetric with respect to its average (e.g. it is usually skewed toward higher values), the normal value is normally slightly lower than the precipitation value at which the curve crosses the 50% point.

The second curve, shown in yellow, labeled "observe d data" , is a probability of exceedance curve derived from the observed data without any model fitting. It steps down every time an observed datum is no longer exceeded. This curve is displayed so that the user may observe how good a fit the smoothed climatological curve is to the actual data. The fitted curves are based on a Gaussian distribution for temperature, and on a gamma distribution for precipitation. The gamma is used for precipitation because of the known asymmetry (skew) of most precipitation distributions, which is fit fairly well by the gamma model but not the symmetric Gaussian model. (More is said about the fitting of precipitation near the bottom of this web site.) The curve based directly on the data is expected to be somewhat rough and irregular, with gaps in some places and clustering in others. This irregularity is caused by the lack of a very long sampling period–i.e., 30 years rather than several hundred! years of observations. If the same number of observations were sampled from an earlier period but the underlying climate remained identical, the places having gaps and clusters would be expected to change. When the irregularities are changeable from one sample to another and have equal chances of appearing in various parts of the distribution, the smooth fitted climatological curve is thought to estimate the true population distribution better than the curve formed from any single sampling of the data. However, in some cases there may be a physical reason for deviations from a smooth distribution. In that case, sampling 500 years of data would not eliminate these features of the curve, these features would be expected to appear somewhat less random (or "noisy" or "jumpy") than features caused purely by sampling variations. For example, a tendency for a plateau of shallow slope might appear near the middle of the "probability of exceedance" distribution, where the steepest slo! pe is usually found, or a steep slope might be found off th! e center of the distribution. The curves derived from the raw data and from the fitted data are presented so that the user may judge whether the fitted curve reasonably describes the raw data. At CPC we believe that in most (but perhaps not all) cases the fitted curve is a better representation of nature than the raw data curve. That is, most of the irregularities in the raw data curve usually occur by chance alone, and would not appear if it were possible to sample a much larger set of cases.

The third curve, shown in red, labeled "final forecast" , represents the probabil ity distribution of the final official CPC forecast. This curve is consistent with the probability anomaly maps that it is designed to accompany, which have been issued since early 1995. That is, the probability printed for the most favored tercile in this new product should correspond to the probability anomaly shown in the probability anomaly maps. When the maps indicate "CL", or climatological probabilities, this product displays a final forecast curve that coincides with the normal curve (and the normal curve hides the final forecast curve). The final forecast curve incorporates all information leading to the forecast, including the recent trend information described by the second curve, but also taking into account the ENSO state, the NAO state (if applicable), and any indications provided by individual forecast tools such as the NCEP or ECHAM dynamical models. The downward slope of the final! forecast curve may be slightly steeper than the slopes of the other two curves, in proportion to the confidence in the final forecast. (Several aspects of the confidence are indicated numerically in several aspects near the top of the graph, to be described below.) This is because when a forecast is known to be particularly skillful (for example, when there is a strong ENSO event in progress and the location is one in which a clear ENSO effect is anticipated), the range of possibilities is smaller than if no useful forecast knowledge were in hand. This represents a decrease in the uncertainty, which shows up as a narrower range of temperature or precipitation values within which the probability of exceedance changes from high to low values. In some cases there is a shift of the forecast curve relative to the normal curve, but without a steeper slope in the forecast curve. This would indicate some confidence in the shift away from the normal, but without a decrease in the rang! e of possibilities in the shifted climate. This might occur! when a trend, or climate change relevant to the present decade as a whole, is believed to be occurring.

A fourth pair of curves, shown by thin red lines, represents and "error envelope" . It is drawn on either side of the main final forecast curve, paralleling the forecast curve. These lines illustrate our estimate of the amount of possible error associated with the forecast curve. The forecast curve itself already conveys great uncertainty about the period being forecast; this is why it is shown in a probabilistic framework and usually has a downward slope that is not much steeper than the slope of the fitted climatological curve. In addition to this inherent uncertainty, there is also some uncertainty related to other factors. Examples of these additional error sources are (1) errors in the most recent observed data used to determine the forecasts, (2) errors the forecasters' perception and judgement of the current climate state, and (3) imperfections in our estimate of our degree of understanding of various climate phenomena such as ENSO, the North Atlantic Oscillation, and lo! ng-term trends. All of these factors could result in some error in the positioning of the forecast curve as a whole. While an accurate evaluation of the size of this error is not possible, an approximation is provided by the error envelope. The approximation is based on the expected sampling variability of the climatological probability of exceedance using 50 years of data, but is amplified to represent the possible misplacement in either direction of the forecast curve as a whole rather than at individual points along the curve. While the basis of the approximation is arbitrary, the resulting envelope is thought to be nonconservative–i.e. the size of the error of the position of the forecast curve is more likely to be over-represented than under-represented. In fact, during most seasons at most U.S. locations the probability anomaly for the favored tercile would need to be 7 to 9 percent in order for the error envelope to exclude the fitted climatological (normal) probability! of exceedance curve over a major portion of the probabilit! y range of the forecast curve. We believe that a large error envelope is justified in order to underscore the need for caution and conservatism on the users' part. While on the subject of caution, it should be noted that while the envelope is smaller at the tails of the distribution than near the middle, it is much larger at the tails in terms of percentage of the difference of the forecast probability from 100% on the left tail, and from 0% on the right tail. This is a reminder that conclusions based on the tails of the forecast curve are dangerous, such as a statement that the chance of being in the 1% tail is 4 times as much as it normally is. The forecast's error envelope shows that quantitative statements about the probability of such an extreme event are not warranted.

Near the bottom of the graph, below the "0%" line on the vertical axis, the observations of the last 10 years (but 15 years for precipitation) are shown by the last two digits of the year. (For example, a "98" would indicate the observation for the year 1998.) These digits provide information about the behavior of the location and season during recent years. The average of the year s shown is indicated by an asterisk on the horizontal "0%" line. The purpose of the display is to give users an idea of how the most recent observations compare with the overall distribution. In some cases the climate of the recent years may tend to be different from the "normal" shown by the entire curve--perhaps mainly lower or mainly higher. Or, they may be more extreme on both sides of the distribution's center than would be expected. If they appear to be unrepresentative of the normal in any respect, the user is faced with the question of whether the climate to occur in the near ! future (i.e. the forecast) will follow the tendency of the recent years. At CPC this issue is examined thoroughly, and is considered as a major component of the forecast. In some cases a strong trend is believed to exist, and is clearly reflected in the forecast, while in other cases the climate of recent years may only be considered a random occurrence. The lowest block of printed information to the right of the curves gives some descriptive information about the tendency of the climate in recent years, including the mean difference from the overall normal.

How to read a "probability of exceedance" curve:

As an example, suppose we first examine the "normal" curve in any one of the graphs. Like the other two curves, the normal curve begins very near the 100% level in the upper left portion of the graph. This indicates that the probability that the temperature or precipitation will exceed the amount shown by the number given at the extreme left of the horizontal axis (below the lower left corner of the graph) is very close to 100%. This makes sense, because the amount has been chosen to be far below the expected normal at the given station and season–an amount that probably has never been observed during the normal base period (except for the occasional case of zero precipitation at certain stations at certain times of the year). This low value is chosen because it is highly unlikely, but not absolutely impossible. The probability that this value will be exceeded is therefore near 100%. As the value is increased from the left toward the right side of the graph, the probability of! it being exceeded begins to decrease, decreasing most rapidly near the middle of the climatological distribution, shown near the middle portion of the graph. The boundaries between the climatologically lowest and middle tercile, and between the middle and highest tercile, are indicated by vertical lines that intercept the normal curve where the probability of exceedance is 66.7% and 33.3%, respectively. A vertical line indicating the median, or near-50% probability of exceedance, is also shown in between the 66.7% and 33.3% vertical lines. (Note: in cases of precipitation, because of skewness, the median and the 50% probability of exceedance do not exactly coincide.) In the right portion of the graph, the "probability of exceedance" line continues to decline and approaches 0% as the horizontal axis values become so large as to be highly unlikely to be exceeded. Because the curve continues to decrease from near 100% to near 0% as the temperature or precipitation value on the h! orizontal axis increases, the probability that the temperat! ure or precipitation will be between any two values on the graph can be determined by subtracting the lower probability of exceedance value from the upper probability of exceedance value. In the case of the "normal" curve, this probability is with respect to the normally expected climatology for the station and season. When this is done for the "last 10 (or 15) years" curve, the probability is with respect to the normally expected climatology that is somewhat modified by the observations that occurred during the most recent 10 (or 15) years, and thus reflects to some extent the effect of the recent trend and its estimated reliability. When the probability is determined with respect to the "final forecast" curve, it is a statement of CPC's official forecast probability. It is useful to compare this probability with that for the normal climatology, and/or for the normal climatology modified by recent trends, to appreciate the difference attributable to the current climate state ! and climate outlook. Because of the modest level of skill in many cases, this difference may often be minor, and, in the case of the "CL" (climatological probability) forecast, there is no difference at all. In some cases there may be a slight difference between the "last 10 (or 15) years" curve and the "normal" curve, but the final forecast curve will coincide with the normal curve. This is the case when the estimated reliability of the recent trend, its anomaly magnitude with respect to the normal curve, or both of these, are believed to be too small to impact the final forecast.

Automatic Probability Evaluation:

In subtracting two "probability of exceedance" values in order to evaluate the probability of occurrence between a lower and upper limit, it is offten difficult to obtain an accurate visual estimate of the probability of exceedance from the graph. For this purpose, the process has been automated for users' convenience in a companion website. The option to use this utility is provided on each graph. Once navigation to that website is accomplished, the user only needs to select a region, a lead time, a variable (temperature or precipitation), and the lower and upper limits within which a probability is to be evaluated. The answer will be computed with respect to all three probability of exceedance curves.

Caution required for the tails of the curves:

Each of the three curves are constructed on the basis of historical observations, and/or the most recent observations, and/or the nature and strength of the impacts of the estimated current climate state. Near the middle of the distribution there has been plentiful data sampled, because the middle of the distribution is most likely and most frequently observed. On the other hand, in the tails, or extremes, of the distribution, there have only been a few cases. Sometimes there may have been no cases in a large portion of a tail, and then just a single observation far out in the extreme part of that tail. Whatever the exact configuration of the observations, the tails are less certain than the middle and the shoulders of the distribution. Therefore, conclusions based on the extreme tails of the distribution are particularly dangerous, and should be made with caution. The probability curves are based on a Gaussian distribution for temperature, and a gamma distribution for precipi! tation. The gamma is used to fit the precipitation data because of the asymmetric, or skewed, nature of precipitation data. In both cases, the shape and length of the tails are based both on the extreme values and on the variability of the values closer to the middle of the distribution. The tails express only an educated guess of the actual extreme value probabilities, and should not be taken literally. A warning about the upper and lower 7% tails of the curves is posted on each graph. The middle 86% of the probability distribution, ranging from the 7% to the 93% probability of exceedance values, is considered to be more reliable than the outer 7% tails. (These cutoffs span the ±1.5 standard deviation interval with respect to the mean, within which adequate numbers of observations are thought to have been sampled.)

Numerical information printed on the graphs:

Above and to the right of the three probability of exceedance curves, selected summary information is printed.

The block of text at the upper left shows the point forecast, or best guess of the numerical forecast. While the point forecast is expressed as an exact number, the certainty that the climate will turn out to be this number is extremely low. That is the whole idea conveyed by the probability of exceedance curves, which always descend slowly to the right, and never suddenly. The gradual rate with which the curves decline implies a large uncertainty. The point forecast is only given to represent the middle, or center, of the forecast distribution, and in no way is intended to imply that we can make a highly accurate numerical forecast. By analogy, when two 6-sided dice are rolled, the midpoint of the distribution of outcomes for the total is 7. However, an outcome of 7, while more likely than any other outcome, is expected to occur only 16.7% of the time on average, if both dice are fair. In the case of our forecasts, the probability of an outcome of exactly (to two decimal plac! es for temperature and precipitation) what our point forecast indicates is very small–usually less than 1%. Again, it is provided to indicate the center of a large distribution of possible outcomes, the size of which is expressed by the probability of exceedance graphs and the computations that can be done on their basis. This warning is very important because our forecasts are highly imperfect , and should never be interpreted as being exact in the way that a predicted time and height of a local high tide at a seaport is exact. Underneath the point forecast, the anomaly forecast , or the departure of the point forecast from the normal, is shown. Beneath the anomaly forecast is the normal, based on the observations during the normal base period as discussed above.

To the right of the above three numbers is more numerical information about the forecast. The percentile (%ile) shows the percentage of cases in the fitted climatological distribution that would be expected to be lower than or equal to the point forecast. For example, if the %ile is 60.0, the point forecast ranks at or above 60.0 percent of the climatological distribution, and less than 40.0 percent of it. In other words, it is a forecast for somewhat, but not highly, above normal conditions. Because of the fairly large degree of uncertainty in the outcome of our forecasts, it is not common for our point forecasts to rank in the top or bottom 20 percent of the climatological distribution. Beneath the %ile of the forecast, the number of standard deviations (#SDs) away from the climatological normal is given for the temperature forecasts. This is a technical measure that is intended for scientific users, and gives exactly the same information as the %ile of the forecast. Every %! ile has a corresponding number of SDs away from the mean. This number is given simply for those who are more used to working with SDs than with %iles. Underneath the number of SDs is the standard deviation (SD) of the climatological distribution for temperature. This is a measure of the variability of the temperature from year to year. Specifically, it gives the amount below and above the normal, forming an interval such that about 68 percent of the cases fall within that interval, assuming a Gaussian distribution. If twice the SD amount is used to form the interval, about 98 percent of the cases would be included. Locations and seasons that have small variations in climate (such as Florida in summer) have a low SD, while places with more year-to-year (or decade-to-decade) variation, such as Montana in winter, have a high SD. Together with the normal (center) number, the SD gives the user an approximate idea of the type of climate normally observed in the given season and loca! tion. [Technical note: The #SDs number equals the anomaly f! orecast divided by the SD.]

On the upper right side of the graph, confidence intervals are given for the forecast. The 50% confidence interval gives two temperature (or precipitation) amounts. The lower amount is the amount that corresponds to the 25%ile (75% probability of exceedance) of the forecast distribution, and the upper amount is the amount that corresponds to the 75%ile (25% probability of exceedance) of the forecast distribution. The amounts between these two limits form an interval that CPC believes has a 50 percent chance of occurring. The 90% confidence interval covers a wider range of amounts, ranging from the 5%ile (95% probability of exceedance) to the 95%ile (5% probability of exceedance) of the forecast. The ranges of amounts covered by the 50% and 90% confidence intervals give an idea of expected error associated with the point forecast. For temperature, the confidence intervals are formed by moving an equal distance on either side of the point forecast. For precipitation, where the ! distribution is usually asymmetric (skewed), the distance upward to the top of the confidence interval is usually greater than the distance downward to the lower boundary of the confidence interval. In either case, the regions outside of the confidence interval limits may be considered to use up the remaining probability equally. The ranges covered by the 50% and 90% confidence intervals are usually fairly wide, in keeping with the uncertainty associated with the point forecast. Note that there is some uncertainty for the confidence intervals themselves, just as there is uncertainty for the probability of exceedance curves themselves (as indicated by the "error envelope"). There is greater expected accuracy for the 50% confidence interval than the 90% confidence interval, because the limits of the 90% interval reach farther into the tails of the forecast distribution.

Below the forecast confidence intervals, are three types of confidence in the forecast . These are estimates of three aspects of the skill expected in the case of this particular forecast. The first, the confidence in shift direction, is a confidence that the climate will deviate from the normal in the direction indicated. The second, the confidence in the point forecast, is confidence that the climate will be close to the point forecast. The third, the integrated confidence, is confidence that the probability distribution as a whole will be different from the climatological distribution. Each of these aspects of confidence is further described next, enabling the user to decide which confidence is more applicable to their particular needs.

Confidence in shift direction: This is a measure of how confident we are that the climate will deviate from the normal in the direction specified, whether below or above. The direction of deviation from the normal is positive when the point forecast exceeds the value of the normal, and negative when it is lower than the normal. The measure, more specifically, is the ratio of the estimated probability that the climate will deviate in the forecast direction to the estimated probability that it will deviate in the opposite direction. That is, it is the odds of the climate deviating in the forecast direction as opposed to it not doing so. For example, if the confidence in the shift direction is 2.00, it indicates that we believe there is twice the probability of a deviation in that direction than in the opposite direction. If the forecast direction is below the normal, a 2.00 confidence means that the probability of below normal conditions is 66.7% and the probability of above nor! mal conditions is 33.3%. If there is no confidence whatsoever regarding which side of the normal will occur, the ratio is exactly 1. Note that the ratio is not that of the probability of the more favored outer tercile to the other, but rather a ratio of the forecast probabilities of occurrence of one half of the climatological distribution to the other half. The dividing line between the two halves is the numerical value of the normal (or center of the distribution) that is posted in the upper left corner of the graph. When the "climatological probabilities" forecast is issued, the shift direction confidence is at its minimum of 1. It may also be 1, however, for a non-"climatological probabilities" forecast--when there is confidence in other aspects of the forecast. One example would be when the likelihood of the near normal category is higher than would be expected climatologically–and the chance of above normal and below normal are both reduced from the climatological chance! . This could occur, for example, in a season and at a locat! ion having high sensitivity to the state of the ENSO, in a case when the ENSO condition is expected to be very close to normal (i.e. neither an El Niño nor a La Niña tendency is expected). In such a case, although the chances of large deviations from normal are reduced, the direction of the shift from normal is just as uncertain as it would be without any knowledge of the ENSO condition. Forecasts for directional shifts may be considered somewhat useful when this confidence measure exceeds 1.5, and more clearly useful when it exceeds 1.8 or even 2 (which is uncommon). It should also be noted that a high confidence in the shift direction usually, but not always, means that the size of the shift is expected to be large. The amount of the expected shift can be seen in the value of the point forecast. In cases with high confidence in the point forecast (another type of confidence, described below), the confidence in the shift direction may be high even though the predicted size of! the shift is only moderate. This is possible because the shift direction refers to any amount of shift in the indicated direction, whether large or small.

Confidence in the point forecast: This is a measure of how narrow, or limited, the distribution of possibilities about the point forecast value is believed to be, compared with the distribution of the historical observations about the normal value. When the forecast distribution is the same, or nearly the same, width as the climatological distribution, this indicates a relative absence of forecast knowledge that would limit the range of possibilities. The measure, more specifically, is the fraction of the width of the forecast distribution to the width of the climatological distribution. When the forecast distribution is no narrower than the observed climatological distribution, the confidence is 1. Confidence values of less than 0.9 are considered moderately helpful, and below 0.8, while rare, are definitely useful. This confidence measure is increased in locations and seasons when climate conditions are known to be related to governing forces (such as ENSO), and the status o! f these forces is able to be somewhat correctly anticipated for the period being forecast. An example of this would be the precipitation during the winter in Florida (and other regions in the southern U.S.), which is partly determined by the ENSO state, and the ENSO state itself is somewhat predictable for forecasts made after the preceding summer. In such a forecast, the possibilities for Florida precipitation are somewhat more limited than they would be with no knowledge of the influence of ENSO or no knowledge of what the ENSO state would likely be during the future period being forecast. This particular confidence measure is not related to the amount of shift of the point forecast from the normal; rather, only the width of the probability distribution about its own central value (the point forecast) is used here. Therefore, forecasts that are close to the normal still may rate high on this confidence measure. Likewise, in some cases there may be a noticeable shift of the p! oint forecast from the normal, but little or no narrowing o! f the distribution. This could occur, for example, when there is a gradual, long-term trend that is used in determining the forecast, but when there is little or no information about differences between the climate this year and the last few years of the same season. In that case, all recent years would be affected by the general trend approximately equally, but their large differences from one another related to factors besides the trend are poorly forecast.

Integrated confidence: an integrated distributional difference from climatology. This is a measure of the estimated totality of all differences between the forecast distribution and the climatological distribution. It includes both distributional shifts and narrowing (i.e. confidence in the point forecast over and above the confidence that would be associated with a climatological forecast), as discussed in the context of the two confidence parameters described above. It would also include distributional deviations of other types that may prove to be possible to predict in the future, such as a widening of the distribution (e.g., as related to an expectation of greater than normal week-to-week variation), or asymmetric or irregular features of the distribution as may be related to specific climate conditions. This measure, specifically, is estimated as the total of the differences in probabilities of exceedance between the climatological distribution and the forecast distribut! ion over the 11 points on the climatological distribution corresponding to its 0.98, 0.90, 0.80, 0.70, ....., 0.20, 0.10, and 0.02 probability of exceedance values. This sum of the differences is then scaled with respect to the result which would be attained when the forecast distribution is completely separated from the climatological distribution. In the case of complete separation, the climatological probability of exceedance remains at 1 (or 100%), or at 0 (or 0%), while the forecast distribution moves through all of its intermediate values. Complete separation, which is unattainable given today's state-of-the-art in climate prediction, would produce a integrated confidence score of 1, while a total absence of separation (as in the case of the "climatological forecast") would produce a score of 0. Integrated confidence values of 0.2 would be considered moderately useful by today's standards, while values of 0.3 would be clearly useful. In examining the integrated confidenc! e values that accompany the graphs, it becomes clear that d! istributional shifts tend to account for the majority of the integrated confidence value, while distribution narrowing contributes to a lesser degree. This characteristic implies that occurrences of extreme climate conditions, whether related to ENSO or to strong decadal trends in progress, represent "forecasts of opportunity", and that forecast skill (and utility) are not constant from year to year for a given location, season and lead time. The only confidence measure (of the three discussed here) that generally remains nearly constant from year to year is the confidence in the point forecast, showing the narrowness of the forecast distribution relative to the climatological distribution. From a practical standpoint, the shift of the forecast distribution from its normal position may often be more important to users than its narrowness. This becomes clearer when one considers the character of the precipitation over a season. If there is a high probability for abnormal wetnes! s, the exact amount of observed precipitation, and its deviation from what was forecast, is often less important to a user than the fact that the precipitation amount was correctly forecast to be above the normal. A forecast of exactly normal precipitation, even with a very narrow forecast distribution, might not represent information as important to the managers of energy companies, or to farmers, as a forecast of deviant precipitation with a much wider probability distribution. It must also be noted that our ability to forecast likely shifts from the normal is currently greater than our ability to narrow the width of the forecast distribution. If, at a distant future time, we become able to significantly narrow the width of the forecast distribution (as we can do currently in 1-day weather forecasts), this would automatically improve our shift direction confidence as well. Our current lack of strong point forecast confidence, fortunately, does not prevent us from having high! shift direction confidence under certain circumstances. Ag! ain, this makes possible "forecasts of opportunity" on an occasional basis.

The middle block of text ion the right side of the graph summarizes the probabilities, for the final forecast, of selected categorical outcomes with respect to the climatological distribution. Included are the probability of the highest 10% of the climatological distribution, the highest third (called "above normal"), the middle third (called "near normal"), the lowest third (called "below normal") and the lowest 10% of the climatological distribution. The probabilities are indicated on the right side of each line of text, and are given to the nearest tenth of one percent. The boundaries of the variable being evaluated (temperature or precipitation) that correspond to these tercile and upper and lower 10% cutoffs, are printed within the parentheses of each text line. Note that the upper and lower 10%, while not completely in the extreme 7% of their tails of the distribution (see the warning above, which is also posted on each graph), are subject to greater uncertainty than the terciles. When the "CL" (clima! tological probability) is given on the traditional map of forecast probability anomaly, the probabilities shown in this block of text will show just that: 10.0%, 33.3%, 33.3%, 33.3%, and 10.0%. When a non-CL forecast is shown on the map, the departure from the climatological probability of the favored tercile in this new product should agree with that given on the map. For example, if there is a 10% probability anomaly for the above normal tercile on the map for the location in question, then the second line of text in this product ("prob of above normal") should be 43.3% (33.3% + 10%). Because a Gaussian fit== is used in the present product, the borrowing rules that are assumed as rough approximations in the probability anomaly maps are not exactly followed. The probability results shown here are considered to be more representative than the approximate "rules of thumb" (with borrowing) assumed for the maps. Thus, we do not always preserve 33.3% probability for the near norma! l tercile when a tilt toward above or below normal is indic! ated, and we do not assume a reduction in the probability of the least favored outer tercile that is equal to the increase in the most favored outer tercile. Additionally, in some cases the middle tercile may have the highest probability anomaly here (by a small amount), even when there is a slight tilt toward one of the two outer terciles on the maps.

The lower block of text pertains to the observations of the most recent 10 years (for temperature) or 15 years (for precipitation). The "norm" refers to the mean of these recent years when there is no skew in the data (e.g. for temperature), and refers to the representative center of the distribution if there is skew (e.g. for most cases of precipitation). The "mean" is the average over the 10 (or 15) most recent years. The median is the center value out of the group of 10 (or 15) years. When there are 10 years, the median is the average of the 5th and 6th highest (also the 6th and 5th lowest) values; when there are 15 years, the median is simply the 8th highest (also 8th lowest) value. The median gives an idea of the middle value, without being affected by the extremeness of the higher and lower values. The "anomaly of the 10 (or 15) year norm" is the norm (given in the line above) minus the overall normal that is given in the upper block of text, discussed above. It indicate! s the departure of the 10 (or 15) year norm from the longer (and older) normal, and in some cases may indicate a systematic trend. Near the bottom of the graph, below the "0%" line on the vertical axis, the observations of the last 10 (or 15) years are shown by the last two digits of the year. These digits are positioned horizontally so that they indicate the temperature or precipitation levels of that year, printed on the horizontal axis just beneath them. An asterisk shows the central value of these observations mean for temperature, median fro precipitation.

How these graphs should NOT be used:

The graphs are to be used at the users' own risk. The probabilities of exceedance are expressions of uncertainty inherent in the climatological distribution and in the final forecast which is conditional on the current climate state. The informed user is aware that in any individual case, the implications of the forecasts may be misleading, as for example when the direction of the shift from the normal turns out to be incorrect. The value of the forecasts is very likely to become visible with repeated use, in which case the frequency of successes will exceed the frequency of failure by an amount that is roughly conveyed by the confidence estimates given with the graphs. This value may or may not show up clearly in individual cases or a small set of cases. To help show how this product should be used, the following are EXAMPLES OF IMPROPER USE OF THIS PRODUCT:
* Treating the "point forecast" as a literal or exact forecast, as in a forecast of tomorrow's maximum temperature. The point forecast is only the center of a wide range of possibilities.
* Using the forecast categorically, without any hedging. The amount of hedging, or weighing and acting on the possibility that the forecast will be incorrect, should be carried out using the probability differences among the alternative outcomes, in conjunction with the costs and the savings associated with each possible sequence of decision and climate outcome. In any individual case, the forecasts should be regarded as probability statements, not as absolute or "all-or-nothing" categorical statements.
* Trusting probability anomalies at face value when they are completely embedded in either of the 7% tails of the distribution. For example: "This year, the chance of a 1-in-100 year drought is 4 times the normal chance." Probability anomalies in the tails should be regarded only as rough estimates.
* Regarding the norm over the last 10 (or 15) years as an indication of the recent trend. In some cases, the departure from normal over the recent years may be due entirely to chance. The confidence score given in the lowest block of text on the graphs can be used to gauge how reliable a recent shift may be considered in reflecting an actual trend.
* Assuming that exactitude in the probabilities, the tercile boundaries, the upper and lower 10% boundaries, or the point forecast or the recent trend implies precision in the forecast itself, or in our knowledge of the "normal" or the recent trend. Precision in any of the quantities presented are only our most precise guesses. The exact probabilities attempt to convey precisely how uncertain we are about the forecasts. THIS PRECISION SHOULD NOT BE INTERPRETED AS IMPLYING FORECAST ACCURACY!

Precipitation Fitting: Precipitation is fitted here using a gamma distribution model. The gamma distribution is thought to be appropriate because it fits the positive skewness of most precipitation distributions. By positive skewness, we mean the tendency for more than half of the cases to fall below the average amount, but for some of the cases that are above the average to be farther above it than any of the below-average cases are below it. The gamma distribution can fit skewed distributions because it has one parameter that fits the general scale of the distribution (e.g. it distinguishes generally wet climates from dry ones, or responds to whether the rainfall is measured in inches or millimeters). Additionally, the gamma distribution has a parameter that fits the shape, or the amount of skewness, in the distribution.
The precipitation forecast distribution is also fit using a gamma model. The parameters of the gamma used for the forecast distribution are determined by the parameters of the climatological distribution in conjunction with the percentile shift implied by the forecast. Both a shift in the scale as well as the shape of the forecast distribution are carried out. This percentile shift is determined the same way for precipitation that it is for temperature, using the probability anomaly of the favored tercile assigned by the forecasters. In meeting that percentile requirement, the gamma parameters are modified such that the forecast distribution becomes less skewed for forecasts of wetter than normal conditions, and more skewed for dry forecasts. As is the case for temperature, the forecast distribution can be viewed as one of the many sub-distributions that make up the climatological distribution. For precipitation, the sub-distributions vary in shape (i.e. skewness) as well as central location, while for temperature they vary in location and width, but not in skewness.
Models besides the gamma have also been used to fit precipitation distributions. For example, a Gaussian model can be used after the data have been subjected to a linear transformation that would eliminate the skewness and make it more Gaussian. One way to carry out a linear transformation would be to raise the original precipitation data to a power that would neutralize the skew.