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Daniel S. Wilks
and
Allan H. Murphy

Abstract

The economic value of current and hypothetically improved seasonal precipitation forecasts is estimated for a regionally important haying/pasturing problem in western Oregon by modeling and analyzing the problem in a decision-analytic framework. Although current forecasts are found to be of relatively little value in this decision-making problem, moderate increases in the quality of the forecasts would lead to substantial increases in their value. The quality/value relationship is sensitive to changes in various economic parameters, including the decision maker's attitude toward risk.

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Allan H. Murphy
and
Harald Daan

Abstract

Subjective probability forecasts of wind speed, visibility and precipitation events for six-hour periods have been prepared on an experimental basis by forecasters at Zierikzec in The Netherlands since October 1980. Results from the first year of the experiment were encouraging, but they revealed a substantial amount of overforecasting (i.e., a strong tendency for forecast probabilities to exceed observed relative frequencies) for all events, periods and forecasters. Moreover, this overforecasting was reflected in a rapid deterioration in the skill of the forecast as a function of lead time. In October 1981 the forecasters were given extensive feedback concerning their individual and collective performance during the first year of the experimental program. The purpose of this paper is to compare the results of the first and second years of the experiment.

Evaluation of the forecasts formulated in the fist and second years of the Zierikzee experiment reveals marked improvements in reliability (i.e., reductions in overforecasting) from year 1 to year 2, both overall and for most stratifications of the results by event, period or forecaster. For example, the reliability of the forecasts increased for all events and periods and for three of the four forecasters. The improvements in reliability are reflected in substantial increases in the skill of the forecasts from year 1 to year 2, with overall skill scores for the second (first) year for the wind speed, visibility and precipitation forecasts of 25.4% (13.9%), 22.4% (12.4%) and 0.5% (−24.7%), respectively. These improvements in performance are attributed to the feedback provided to the forecasters at the beginning of the second year of the experiment and to the experience in probability forecasting gained by the forecasters during the first year of the program.

The paper concludes with a brief discussion of the results and their implications for probability forecasting in meteorology.

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Seijo Kruizinga
and
Allan H. Murphy

Abstract

The purpose of this paper is to describe some results of a study in which an analogue procedure developed in The Netherlands is used to formulate objective probabilistic temperature forecasts on an experimental basis. As currently employed, the procedure routinely provides forecasters at the Royal Netherlands Meteorological Institute with guidance information that summarizes, for days 11 through 6, the weather conditions associated with the best thirty analogues of the corresponding forecast situation. In the work reported here, the empirical frequency distribution of maximum temperature corresponding to these thirty analogues is used to generate both categorical and probabilistic forecasts of this element. Attention is focused on three types of probabilistic forecasts of maximum temperature: 1) a discrete distribution for five temperature classes 2) a variable-width credible interval., and 3) a fixed-width credible interval.

Results of the experiment indicate that all three types of probabilistic temperature forecasts are quite reliable, in the sense that the forecast probabilities correspond closely to the relative frequencies of observed temperatures associated with these class/intervals. Moreover, the forecasts generally are more accurate and precise, according to several different measures of performance, than forecasts based on standards of reference such as climatology and persistence. Thus, these experimental objective probability forecasts usually exhibit positive skill. As expected, the level of skill decreases markedly from day 1 to day 6 for all three types of probability forecasts. Evaluation of two types of categorical forecasts—the median and mean temperatures derived from the empirical frequency distribution—reveals similar results.

The implications of the results of this study for operational temperature forecasting are discussed briefly, and some possible refinements and/or improvements in objective probabilistic temperature forecasting are described.

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Allan H. Murphy
and
Qian Ye

Abstract

A time-dependent version of the cost-loss ratio situation is described and the optimal use and economic value of meteorological information are investigated in this decision-making problem. The time-dependent situation is motivated by a decision maker who contemplates postponing the protect/do not protect decision in anticipation of obtaining more accurate forecasts at some later time (i.e., shorter lead time), but who also recognizes that the cost of protection will increase as lead time decreases. Imperfect categorical forecasts, calibrated according to past performance, constitute the information of primary interest. Optimal decisions are based on minimizing expected expense and the value of information is measured relative to the expected expense associated with climatological information.

Accuracy and cost of protection are modeled as exponentially decreasing functions of lead time, and time-dependent expressions for expected expense and value of information are derived. An optimal lead time is identified that corresponds to the time at which the expected expense associated with imperfect forecasts attains its minimum value. The effects of the values of the parameters in the accuracy and cost-of-protection models on expected expense, optimal lead time, and forecast value are examined. Moreover, the optimal lead time is shown to differ in some cases from the lead time at which the economic value of imperfect forecasts is maximized. Numerical examples are presented to illustrate the various results. The implications of these results are discussed and some possible extensions of this work are suggested.

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Allan H. Murphy
and
Qian Ye

Abstract

In this paper the sufficiency relation is used to compare objective and subjective probability of precipitation (PoP) forecasts. The theoretical significance of the sufficiency relation in comparative evaluation arises from the fact that if it can be shown that forecasting system A is sufficient for forecasting system B, then A's forecasts are necessarily of higher quality and greater value to all users than B's forecasts. However, since the sufficiency relation is an incomplete order (it is not always possible to show that system A is sufficient for system B, or vice versa), the practical significance of this relation warrants further investigation.

An operational method of comparing forecasting systems using the sufficiency relation has recently been described in the forecasting literature. This method involves the construction of a so-called forecast sufficiency characteristic (FSC) for each forecasting system, based on a representative set of forecasts and observations. In terms of this characterization, system A is sufficient for system B if A's FSC is superior to B's FSC.

Objective and subjective PoP forecasts for six National Weather Service offices are compared here in terms of their respective FSCs. Sufficiency was found in only two of the 24 cases defined by various combinations of forecast office, season, and lead time. In these two cases, both involving 12–24 hour forecasts, the subjective forecasts were sufficient for the objective forecasts. Several other cases exhibited a condition described as "almost sufficient,” but caution must be exercised in drawing conclusions regarding the relative quality and/or relative value of forecasts in such cases. Comparison of the FSCs of PoP forecasts and the FSCs of categorical forecasts derived from the PoP forecasts reveals that, as expected, the former are always superior to the latter. The implications of these results for comparative evaluation of weather forecasting systems are discussed and some possible extensions of this work are identified.

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Robert L. Winkler
and
Allan H. Murphy

Abstract

An experiment was conducted at the National Weather Service Forecast Office in St. Louis, Mo., to investigate the ability of forecasters to differentiate among different points in a forecast area with regard to the livelihood of the occurrence of measurable precipitation and the relative ability of forecasters to make point and area precipitation probability forecasts. On each forecasting occasion in the experimental period (November 1972–March 1973), the forecasters made an average point probability forecast for the St. Louis metropolitan area, point probability forecasts for five specific points in the area, an area probability forecast, and an expected areal coverage forecast.

The results indicate that the forecasters did not differentiate among the five points very often, but that this absence of differences among the point probabilities was justified by the lack of variability exhibited by the observations of precipitation occurrence at these points during the experimental period. Evaluations of the average point probability forecasts, individual point probability forecasts, and expected areal coverage forecasts reveal that these forecasts were quite reliable and accurate and that they were also internally consistent. The area probability forecasts, however, tended not to be consistent with the other forecasts, and the average area probability forecast was considerably lower than the relative frequency of occurrence of precipitation “somewhere in the area.”

The implications of these results for precipitation probability forecasting in meteorology are briefly discussed.

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Allan H. Murphy
and
Robert L. Winkler

Abstract

This paper describes the results of an experiment involving credible interval temperature forecasts. A credible interval is an interval of values of the variable of concern, in this case maximum or minimum temperature, accompanied by a probability which expresses a forecaster's “degree of belief” that the temperature will fall in the given interval. The experiment was designed to investigate the ability of fore-casters to express the uncertainty inherent in their temperature forecasts in probabilistic terms and to compare two approaches (variable-width and flied-width intervals) to credible interval temperature forecasting.

Four experienced weather forecasters participated in the experiment, which was conducted at the National Weather Service Forecast Office in Denver, Colorado. Two forecasters made variable-width, fixed-probability forecasts using 50% and 75% intervals, while the other two forecasters made fixed-width, variable-probability forecasts using 5°F and 9°F intervals. On each occasion the forecasters first determined a median, and the variable-width and fixed-width intervals were then centered at the median in terms of probability and width, respectively.

The results indicate that, overall, the medians determined by the forecasters were good point forecasts of maximum and minimum temperatures. Further, a comparison of the average errors for the forecasters’ medians with the average errors for the medians derived from climatology reveals that the forecasters were able to improve greatly upon climatology. The variable-width credible intervals were very reliable in the sense that the observed relative frequencies corresponded very closely to the forecast probabilities. Moreover, the variable-width intervals were more reliable and much more precise than the corresponding climatological forecasts. The fixed-width intervals, on the other hand, were assigned probabilities that were, on the average, larger that the corresponding relative frequencies.

In summary, the results indicate that weather forecasters can use credible intervals to describe the uncertainty contained in their temperature forecasts. The implications of these experimental results for probability forecasting in general and temperature forecasting in particular are discussed.

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Allan H. Murphy
and
Jack C. Thompson

Abstract

t has been shown previously that ordinal relationships between measures of the accuracy and value ofprobability forecasts do not exist, in general, in N-state (N > 2) situations. Some implications of this resultare illustrated by comparing the accuracy and value of such forecasts in a realistic decision-making situation-a three-action, three-state situation involving the protection of a fruit orchard against frosts and freezes.Geometrical interpretations of the forecasts and measures are described and then used to investigate the existence of ordinal relationships in this so-called fruit-frost situation. The results indicate, as expected, thatan increase in forecast accuracy can lead to a decrease in forecast value. Some generalizations and speculations related to the existence and nonexistence of such ordinal relationships are presented.

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Allan H. Murphy
and
Robert L. Winkler

Abstract

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Edward S. Epstein
and
Allan H. Murphy

Abstract

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